Method for managing production from a hydrocarbon producing reservoir in real-time

ABSTRACT

The invention relates to a method of performing an oilfield operation of an oilfield having at least one wellsite, each wellsite having a wellbore penetrating a subterranean formation for extracting fluid from an underground reservoir therein. The method steps include obtaining a plurality of real-time parameters from a plurality of sensors disposed about the oilfield, wherein the plurality of real-time parameters comprise at least one selected from a group consisting of real-time flow rate data and real-time pressure data of the wellbore, configuring a gridless analytical simulator for simulating the underground reservoir based on the plurality of real-time parameters, generating real-time simulation results of the underground reservoir and the at least one wellsite in real-time using the gridless analytical simulator, and performing the oilfield operation based on the real-time simulation results

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims priority under 35 U.S.C. § 119(e) fromProvisional Patent Application Nos. 60/953,449 filed Aug. 1, 2007 withAttorney Docket No. 09469/115001; 110.158-PRV1, 60/956,070 filed Aug.15, 2007 with Attorney Docket No. 09469/115002; 110.158-PRV2, and61/027,801 filed Feb. 11, 2008 with Attorney Docket No. 09469/115003;110.158-PRV3. This is a continuation-in-part (CIP) application of andclaims priority under 35 U.S.C. § 120 to U.S. patent application Ser.No. 11/924,560 filed on Oct. 25, 2007 with Attorney Docket No.09469/024003; 94.0073.

BACKGROUND

1. Field of the Invention

The present invention relates to techniques for performing oilfieldoperations relating to subterranean formations having reservoirstherein. More particularly, the invention relates to techniques forperforming oilfield operations involving an analysis of reservoiroperations, and their impact on such oilfield operations.

2. Background of the Related Art

Oilfield operations, such as surveying, drilling, wireline testing,completions, production, planning and oilfield analysis, are typicallyperformed to locate and gather valuable downhole fluids. Various aspectsof the oilfield and its related operations are shown in FIGS. 1A-1D. Asshown in FIG. 1A, surveys are often performed using acquisitionmethodologies, such as seismic scanners to generate maps of undergroundstructures. These structures are often analyzed to determine thepresence of subterranean assets, such as valuable fluids or minerals.This information is used to assess the underground structures and locatethe formations containing the desired subterranean assets. Datacollected from the acquisition methodologies may be evaluated andanalyzed to determine whether such valuable items are present, and ifthey are reasonably accessible.

As shown in FIG. 1B-1D, one or more wellsites may be positioned alongthe underground structures to gather valuable fluids from thesubterranean reservoirs. The wellsites are provided with tools capableof locating and removing hydrocarbons from the subterranean reservoirs.As shown in FIG. 1B, drilling tools are typically advanced from the oilrigs and into the earth along a given path to locate the valuabledownhole fluids. During the drilling operation, the drilling tool mayperform downhole measurements to investigate downhole conditions. Insome cases, as shown in FIG. 1C, the drilling tool is removed and awireline tool is deployed into the wellbore to perform additionaldownhole testing.

After the drilling operation is complete, the well may then be preparedfor production. As shown in FIG. 1D, wellbore completions equipment isdeployed into the wellbore to complete the well in preparation for theproduction of fluid therethrough. Fluid is then drawn from downholereservoirs, into the wellbore and flows to the surface. Productionfacilities are positioned at surface locations to collect thehydrocarbons from the wellsite(s). Fluid drawn from the subterraneanreservoir(s) passes to the production facilities via transportmechanisms, such as tubing. Various equipment may be positioned aboutthe oilfield to monitor oilfield parameters and/or to manipulate theoilfield operations.

During the oilfield operations, data is typically collected for analysisand/or monitoring of the oilfield operations. Such data may include, forexample, subterranean formation, equipment, historical and/or otherdata. Data concerning the subterranean formation is collected using avariety of sources. Such formation data may be static or dynamic. Staticdata relates to, for example, formation structure and geologicalstratigraphy that define the geological structure of the subterraneanformation. Dynamic data relates to, for example, fluids flowing throughthe geologic structures of the subterranean formation over time. Suchstatic and/or dynamic data may be collected to learn more about theformations and the valuable assets contained therein.

Sources used to collect static data may be seismic tools, such as aseismic truck that sends compression waves into the earth as shown inFIG. 1A. These waves are measured to characterize changes in the densityof the geological structure at different depths. This information may beused to generate basic structural maps of the subterranean formation.Other static measurements may be gathered using core sampling and welllogging techniques. Core samples may be used to take physical specimensof the formation at various depths as shown in FIG. 1B. Well loggingtypically involves deployment of a downhole tool into the wellbore tocollect various downhole measurements, such as density, resistivity,etc., at various depths. Such well logging may be performed using, forexample, the drilling tool of FIG. 1B and/or the wireline tool of FIG.1C. Once the well is formed and completed, fluid flows to the surfaceusing production tubing as shown in FIG. 1D. As fluid passes to thesurface, various dynamic measurements, such as fluid flow rates,pressure, and composition may be monitored. These parameters may be usedto determine various characteristics of the subterranean formation.

Sensors may be positioned about the oilfield to collect data relating tovarious oilfield operations. For example, sensors in the drillingequipment may monitor drilling conditions, sensors in the wellbore maymonitor fluid composition, sensors located along the flow path maymonitor flow rates, and sensors at the processing facility may monitorfluids collected. Other sensors may be provided to monitor downhole,surface, equipment or other conditions.

The monitored data is often used to make decisions at various locationsof the oilfield at various times. Data collected by these sensors may befurther analyzed and processed. Data may be collected and used forcurrent or future operations. When used for future operations at thesame or other locations, such data may sometimes be referred to ashistorical data.

The processed data may be used to predict downhole conditions, and makedecisions concerning oilfield operations. Such decisions may involvewell planning, well targeting, well completions, operating levels,production rates and other operations and/or conditions. Often thisinformation is used to determine when to drill new wells, re-completeexisting wells, or alter wellbore production.

Data from one or more wellbores may be analyzed to plan or predictvarious outcomes at a given wellbore. In some cases, the data fromneighboring wellbores or wellbores with similar conditions or equipmentmay be used to predict how a well will perform. There are usually alarge number of variables and large quantities of data to consider inanalyzing oilfield operations. It is, therefore, often useful to modelthe behavior of the oilfield operation to determine the desired courseof action. During the ongoing operations, the operating conditions mayneed adjustment as conditions change and new information is received.

Techniques have been developed to model the behavior of various aspectsof the oilfield operations, such as geological structures, downholereservoirs, wellbores, surface facilities as well as other portions ofthe oilfield operation. Typically, there are different types ofsimulators for different purposes. For example, there are simulatorsthat focus on reservoir properties, wellbore production, or surfaceprocessing. Examples of simulators that may be used at the wellsite aredescribed in U.S. Pat. No. 5,992,519 and WO2004049216. Other examples ofthese modeling techniques are shown in U.S. Pat. Nos. 5,992,519,6,313,837, WO1999/064896, WO2005/122001, US2003/0216897, US2003/0132934,US2005/0149307, and US2006/0197759.

Recent attempts have been made to consider a broader range of data inoilfield operations. For example, U.S. Pat. No. 6,842,700 to Poedescribes a method for evaluating a well and a reservoir without theneed for well pressure history. In another example, US2006/0069511 toThambynayagam discloses a gas reservoir evaluation and assessment tool.Other examples of such recent attempts are disclosed in U.S. Pat. Nos.6,018,497, 6,078,869, 6,106,561, 6,230,101, 6,980,940, 7,164,990,GB2336008, US2004/0220846, US2006/0129366, US2006/0184329, U.S. Ser. No.10/586,283, and WO04049216.

Despite the development and advancement of wellbore modeling and/orsimulation techniques, many of which employ finite difference numericalmethods to construct reservoir models, there remains a need to providetechniques capable of performing real-time simulations for the oilfieldoperation. It would be desirable to have a system that performssimulations that consider data throughout the oilfield operation. Insome cases, it may be desirable to continuously monitor and analyzeoilfield data, anticipate and identify events, and to perform real-timediagnostics and interpretation of the oilfield data. In other cases, itmay be desirable to support real-time decision making for performingoilfield operations. It is further desirable that such techniques becapable of one of more of the following, among others: taking intoconsideration the effects of production from other wells in the samereservoir; updating the reservoir model based on history matching; andautomatic workflow with real-time plotting of key parameters againsttime and real-time alarms based on pre-determined criteria.

SUMMARY

In general, in one aspect, the invention relates to a method ofperforming an oilfield operation of an oilfield having at least onewellsite, each wellsite having a wellbore penetrating a subterraneanformation for extracting fluid from an underground reservoir therein.The method steps include obtaining a plurality of real-time parametersfrom a plurality of sensors disposed about the oilfield, wherein theplurality of real-time parameters comprise at least one selected from agroup consisting of real-time flow rate data and real-time pressure dataof the wellbore, configuring a gridless analytical simulator forsimulating the underground reservoir based on the plurality of real-timeparameters, generating real-time simulation results of the undergroundreservoir and the at least one wellsite in real-time using the gridlessanalytical simulator, and performing the oilfield operation based on thereal-time simulation results.

In general, in one aspect, the invention relates to a method ofperforming an oilfield operation of an oilfield having a plurality ofwellsites, each wellsite having a wellbore penetrating a subterraneanformation for extracting fluid from an underground reservoir therein.The method steps include obtaining real-time pressure data from apermanent down-hole pressure gauge, identifying a reservoir model for agridless analytical simulator based on a rate of change of the real-timepressure data using a neural network method, generating real-timesimulation results of the underground reservoir and the plurality ofwellsites in real-time using the gridless analytical simulator, andperforming the oilfield operation based on the real-time simulationresults.

In general, in one aspect, the invention relates to a method ofperforming an oilfield operation of an oilfield having a plurality ofgas wells, each gas well having a wellbore penetrating a subterraneanformation for extracting gas from an underground reservoir therein. themethod steps include obtaining real-time flow rate data from a flowmeter, obtaining at least one selected from a group consisting ofreal-time pressure data and offline pressure data, generating a firstsimulation result of the underground reservoir and the plurality of gaswells using a non-linear regression model with the real-time flow ratedata, and the real-time pressure data, and the offline pressure data ifthe real-time pressure data is not available, identifying a reservoirmodel for a gridless analytical simulator using a neural network methodif the real-time pressure data is available, generating a secondsimulation result of the reservoir and the plurality of gas wells inreal-time using the gridless analytical simulator, and performing theoilfield operation based on at least one selected from a groupconsisting of the first simulation result and the second simulationresult.

In general, in one aspect, the invention relates to a computer readablemedium, embodying instructions executable by a computer to performmethod steps for an oilfield operation, the oilfield having at least onewellsite, each of the at least one wellsite having a wellborepenetrating a subterranean formation for extracting fluid from anunderground reservoir therein. The instructions include functionality toobtain a plurality of real-time parameters from a plurality of sensorsdisposed about the oilfield, wherein the plurality of real-timeparameters comprise at least one selected from a group consisting offlow rate and pressure of the wellbore, configure a gridless analyticalsimulator for simulating the reservoir based on the plurality ofreal-time parameters, and generate real-time simulation results of thereservoir and the at least one wellsite in real-time using the gridlessanalytical simulator, wherein the oilfield operation is performed basedon the real-time simulation results.

In general, in one aspect, the invention relates to a computer readablemedium, embodying instructions executable by a computer to performmethod steps for an oilfield operation, the oilfield having a pluralityof wellsites, each of the plurality of wellsites having a wellborepenetrating a subterranean formation for extracting fluid from anunderground reservoir therein. The instructions include functionality toobtain real-time pressure data from a permanent down-hole pressuregauge, identify a reservoir model for a gridless analytical simulatorbased on a rate of change of the real-time pressure data using a neuralnetwork method, generate real-time simulation results of the reservoirand the plurality of wellsites in real-time using the gridlessanalytical simulator, and perform the oilfield operation based on thereal-time simulation results.

In general, in one aspect, the invention relates to a computer readablemedium, embodying instructions executable by a computer to performmethod steps for an oilfield operation, the oilfield having a pluralityof gas wells, each of the plurality of gas wells having a wellborepenetrating a subterranean formation for extracting gas from anunderground reservoir therein. The instructions include functionality toobtain real-time flow rate data from a flow meter, obtain at least oneselected from a group consisting of real-time pressure data and offlinepressure data, generate a first simulation result of the undergroundreservoir and the plurality of gas wells using a non-linear regressionmodel with the real-time flow rate data, and the real-time pressuredata, and the offline pressure data if the real-time pressure data isnot available, identify a reservoir model for a gridless analyticalsimulator using a neural network method if the real-time pressure datais available, generate a second simulation result of the reservoir andthe plurality of gas wells in real-time using the gridless analyticalsimulator, and perform the oilfield operation based on at least oneselected from a group consisting of the first simulation result and thesecond simulation result.

Other aspects and advantages of the invention will be apparent from thefollowing description and the appended claims.

BRIEF DESCRIPTION OF THE DRAWINGS

So that the above recited features and advantages of the presentinvention can be understood in detail, a more particular description ofthe invention, briefly summarized above, may be had by reference to theembodiments thereof that are illustrated in the appended drawings. It isto be noted, however, that the appended drawings illustrate only typicalembodiments of this invention and are therefore not to be consideredlimiting of its scope, for the invention may admit to other equallyeffective embodiments.

FIGS. 1A-1D show exemplary schematic views of an oilfield havingsubterranean structures including reservoirs therein and variousoilfield operations being performed on the oilfield. FIG. 1A depicts anexemplary survey operation being performed by a seismic truck. FIG. 1Bdepicts an exemplary drilling operation being performed by a drillingtool suspended by a rig and advanced into the subterranean formation.FIG. 1C depicts an exemplary wireline operation being performed by awireline tool suspended by the rig and into the wellbore of FIG. 1B.FIG. 1D depicts an exemplary production operation being performed by aproduction tool being deployed from the rig and into a completedwellbore for drawing fluid from the downhole reservoir into a surfacefacility.

FIGS. 2A-2D are exemplary graphical depictions of data collected by thetools of FIGS. 1A-1D, respectively. FIG. 2A depicts an exemplary seismictrace of the subterranean formation of FIG. 1A. FIG. 2B depictsexemplary core sample of the formation shown in FIG. 1B. FIG. 2C depictsan exemplary well log of the subterranean formation of FIG. 1C. FIG. 2Ddepicts an exemplary production decline curve of fluid flowing throughthe subterranean formation of FIG. 1D.

FIG. 3 shows an exemplary schematic view, partially in cross section, ofan oilfield having a plurality of data acquisition tools positioned atvarious locations along the oilfield for collecting data from thesubterranean formation.

FIG. 4 shows an exemplary schematic view of an oilfield having aplurality of wellsites for producing hydrocarbons from the subterraneanformation.

FIG. 5 shows an exemplary schematic diagram of a portion of the oilfieldof FIG. 4 depicting the production operation in detail.

FIG. 6 is a flow chart of a permanent downhole pressure gauge (PDG)workflow in an oilfield.

FIG. 7 is a flow chart of a gas rate workflow in a gas field.

FIG. 8 shows an exemplary schematic diagram of a reservoir modeled in agridless analytical simulator.

FIG. 9 is a flow chart of a method to perform an oilfield operationusing the real-time analytical simulator.

DETAILED DESCRIPTION

Presently preferred embodiments of the invention are shown in theabove-identified figures and described in detail below. In describingthe preferred embodiments, like or identical reference numerals are usedto identify common or similar elements. The figures are not necessarilyto scale and certain features and certain views of the figures may beshown exaggerated in scale or in schematic in the interest of clarityand conciseness.

FIGS. 1A-D show an oilfield (100) having geological structures and/orsubterranean formations therein. As shown in these figures, variousmeasurements of the subterranean formation are taken by different toolsat the same location. These measurements may be used to generateinformation about the formation and/or the geological structures and/orfluids contained therein.

FIGS. 1A-1D depict schematic views of an oilfield (100) havingsubterranean formations (102) containing a reservoir (104) therein anddepicting various oilfield operations being performed on the oilfield(100). FIG. 1A depicts a survey operation being performed by a seismictruck (106 a) to measure properties of the subterranean formation. Thesurvey operation is a seismic survey operation for producing soundvibration(s) (112). In FIG. 1A, one such sound vibration (112) isgenerated by a source (110) and reflects off a plurality of horizons(114) in an earth formation (116). The sound vibration(s) (112) is (are)received in by sensors (S), such as geophone-receivers (118), situatedon the earth's surface, and the geophone-receivers (118) produceelectrical output signals, referred to as data received (120) in FIG. 1.

In response to the received sound vibration(s) (112) representative ofdifferent parameters (such as amplitude and/or frequency) of the soundvibration(s) (112). The data received (120) is provided as input data toa computer (122 a) of the seismic recording truck (106 a), andresponsive to the input data, the recording truck computer (122 a)generates a seismic data output record (124). The seismic data may befurther processed as desired, for example by data reduction.

FIG. 1B depicts a drilling operation being performed by a drilling tool(106 b) suspended by a rig (128) and advanced into the subterraneanformation (102) to form a wellbore (136). A mud pit (130) is used todraw drilling mud into the drilling tool (106 b) via flow line (132) forcirculating drilling mud through the drilling tool (106 b) and back tothe surface. The drilling tool (106 b) is advanced into the formation toreach reservoir (104). The drilling tool (106 b) is preferably adaptedfor measuring downhole properties. The drilling tool (106 b) may also beadapted for taking a core sample (133), as shown, or removed so that acore sample (133) may be taken using another tool.

A surface unit (134) is used to communicate with the drilling tool (106b) and offsite operations. The surface unit (134) is capable ofcommunicating with the drilling tool (106 b) to send commands to drivethe drilling tool (106 b), and to receive data therefrom. The surfaceunit (134) is preferably provided with computer facilities forreceiving, storing, processing, and analyzing data from the oilfield(100). The surface unit (134) collects data output (135) generatedduring the drilling operation. Computer facilities, such as those of thesurface unit (134), may be positioned at various locations about theoilfield (100) and/or at remote locations.

Sensors (S), such as gauges, may be positioned throughout the reservoir,rig, oilfield equipment (such as the downhole tool), or other portionsof the oilfield for gathering information about various parameters, suchas surface parameters, downhole parameters, and/or operating conditions.These sensors (S) preferably measure oilfield parameters, such as weighton bit, torque on bit, pressures, temperatures, flow rates, compositionsand other parameters of the oilfield operation.

The information gathered by the sensors (S) may be collected by thesurface unit (134) and/or other data collection sources for analysis orother processing. The data collected by the sensors (S) may be usedalone or in combination with other data. The data may be collected in adatabase and all or select portions of the data may be selectively usedfor analyzing and/or predicting oilfield operations of the currentand/or other wellbores.

Data outputs from the various sensors (S) positioned about the oilfieldmay be processed for use. The data may be historical data, real timedata, or combinations thereof. The real time data may be used in realtime, or stored for later use. The data may also be combined withhistorical data or other inputs for further analysis. The data may behoused in separate databases, or combined into a single database.

The collected data may be used to perform analysis, such as modelingoperations. For example, the seismic data output may be used to performgeological, geophysical, reservoir engineering, and/or productionsimulations. The reservoir, wellbore, surface and/or process data may beused to perform reservoir, wellbore, or other production simulations.The data outputs from the oilfield operation may be generated directlyfrom the sensors (S), or after some preprocessing or modeling. Thesedata outputs may act as inputs for further analysis.

The data is collected and stored at the surface unit (134). One or moresurface units (134) may be located at the oilfield (100), or linkedremotely thereto. The surface unit (134) may be a single unit, or acomplex network of units used to perform the necessary data managementfunctions throughout the oilfield (100). The surface unit (134) may be amanual or automatic system. The surface unit (134) may be operatedand/or adjusted by a user.

The surface unit (134) may be provided with a transceiver (137) to allowcommunications between the surface unit (134) and various portions (orregions) of the oilfield (100) or other locations. The surface unit(134) may also be provided with or functionally linked to a controllerfor actuating mechanisms at the oilfield (100). The surface unit (134)may then send command signals to the oilfield (100) in response to datareceived. The surface unit (134) may receive commands via thetransceiver or may itself execute commands to the controller. Aprocessor may be provided to analyze the data (locally or remotely) andmake the decisions to actuate the controller. In this manner, theoilfield (100) may be selectively adjusted based on the data collectedto optimize fluid recovery rates, or to maximize the longevity of thereservoir and its ultimate production capacity. These adjustments may bemade automatically based on computer protocol, or manually by anoperator. In some cases, well plans may be adjusted to select optimumoperating conditions, or to avoid problems.

FIG. 1C depicts a wireline operation being performed by a wireline tool(106 c) suspended by the rig (128) and into the wellbore (136) of FIG.1B. The wireline tool (106 c) is preferably adapted for deployment intoa wellbore (136) for performing well logs, performing downhole testsand/or collecting samples. The wireline tool (106 c) may be used toprovide another method and apparatus for performing a seismic surveyoperation. The wireline tool (106 c) of FIG. 1C may have an explosive oracoustic energy source (143) that provides electrical signals to thesurrounding subterranean formations (102).

The wireline tool (106 c) may be operatively linked to, for example, thegeophones (118) stored in the computer (122 a) of the seismic recordingtruck (106 a) of FIG. 1A. The wireline tool (106 c) may also providedata to the surface unit (134). As shown, data output (135) is generatedby the wireline tool (106 c) and collected at the surface. The wirelinetool (106 c) may be positioned at various depths in the wellbore (136)to provide a survey of the subterranean formation.

FIG. 1D depicts a production operation being performed by a productiontool (106 d) deployed from a production unit or christmas tree (129) andinto the completed wellbore (136) of FIG. 1C for drawing fluid from thedownhole reservoirs into the surface facilities (142). Fluid flows fromreservoir (104) through perforations in the casing (not shown) and intothe production tool (106 d) in the wellbore (136) and to the surfacefacilities (142) via a gathering network (146).

Sensors (S), such as gauges, may be positioned about the oilfield tocollect data relating to various oilfield operations as describedpreviously. As shown, the sensor (S) may be positioned in the productiontool (106 d) or associated equipment, such as the Christmas tree,gathering network, surface facilities and/or the production facility, tomeasure fluid parameters, such as fluid composition, flow rates,pressures, temperatures, and/or other parameters of the productionoperation.

While only simplified wellsite configurations are shown, it will beappreciated that the oilfield may cover a portion of land, sea, and/orwater locations that hosts one or more wellsites. Production may alsoinclude injection wells (not shown) for added recovery. One or moregathering facilities may be operatively connected to one or more of thewellsites for selectively collecting downhole fluids from thewellsite(s).

During the production process, data output (135) may be collected fromvarious sensors (S) and passed to the surface unit (134) and/orprocessing facilities. This data may be, for example, reservoir data,wellbore data, surface data, and/or process data.

While FIGS. 1A-1D depict monitoring tools used to measure properties ofan oilfield (100), it will be appreciated that the tools may be used inconnection with non-oilfield operations, such as mines, aquifers orother subterranean facilities. Also, while certain data acquisitiontools are depicted, it will be appreciated that various measurementtools capable of sensing properties, such as seismic two-way traveltime, density, resistivity, production rate, etc., of the subterraneanformation and/or its geological structures may be used. Various sensors(S) may be located at various positions along the subterranean formationand/or the monitoring tools to collect and/or monitor the desired data.Other sources of data may also be provided from offsite locations.

The oilfield configuration in FIGS. 1A-1D is not intended to limit thescope of the invention. Part, or all, of the oilfield (100) may be onland and/or sea. Also, while a single oilfield at a single location isdepicted, the present invention may be used with any combination of oneor more oilfields (100), one or more processing facilities and one ormore wellsites. Additionally, while only one wellsite is shown, it willbe appreciated that the oilfield (100) may cover a portion of land thathosts one or more wellsites. One or more gathering facilities may beoperatively connected to one or more of the wellsites for selectivelycollecting downhole fluids from the wellsite(s).

FIGS. 2A-2D are graphical depictions of data collected by the tools ofFIGS. 1A-D, respectively. FIG. 2A depicts a seismic trace (202) of thesubterranean formation of FIG. 1A taken by survey tool (106 a). Theseismic trace measures a two-way response over a period of time. FIG. 2Bdepicts a core sample (133) taken by the drilling tool (106 b). The coretest typically provides a graph of the density, resistivity, or otherphysical property of the core sample (133) over the length of the core.Tests for density and viscosity are often performed on the fluids in thecore at varying pressures and temperatures. FIG. 2C depicts a well log(204) of the subterranean formation of FIG. 1C taken by the wirelinetool (106 c). The wireline log typically provides a resistivitymeasurement of the formation at various depths. FIG. 2D depicts aproduction decline curve (206) of fluid flowing through the subterraneanformation of FIG. 1D taken by the production tool (106 d). Theproduction decline curve (206) typically provides the production rate Qas a function of time t.

The respective graphs of FIGS. 2A-2C contain static measurements thatdescribe the physical characteristics of the formation. Thesemeasurements may be compared to determine the accuracy of themeasurements and/or for checking for errors. In this manner, the plotsof each of the respective measurements may be aligned and scaled forcomparison and verification of the properties.

FIG. 2D provides a dynamic measurement of the fluid properties throughthe wellbore. As the fluid flows through the wellbore, measurements aretaken of fluid properties, such as flow rates, pressures, composition,etc. As described below, the static and dynamic measurements may be usedto generate models of the subterranean formation to determinecharacteristics thereof.

FIG. 3 is a schematic view, partially in cross section of an oilfield(300) having data acquisition tools (302 a), (302 b), (302 c), and (302d) positioned at various locations along the oilfield for collectingdata of a subterranean formation (304). The data acquisition tools (302a-302 d) may be the same as data acquisition tools (106 a-106 d) of FIG.1, respectively. As shown, the data acquisition tools (302 a-302 d)generate data plots or measurements (308 a-308 d), respectively.

Data plots (308 a-308 c) are examples of static data plots that may begenerated by the data acquisition tools (302 a-302 d), respectively.Static data plot (308 a) is a seismic two-way response time and may bethe same as the seismic trace (202) of FIG. 2A. Static plot (308 b) iscore sample data measured from a core sample of the formation (304),similar to the core sample (133) of FIG. 2B. Static data plot (308 c) isa logging trace, similar to the well log (204) of FIG. 2C. Data plot(308 d) is a dynamic data plot of the fluid flow rate over time, similarto the graph (206) of FIG. 2D. Other data may also be collected, such ashistorical data, user inputs, economic information, other measurementdata, and other parameters of interest.

The subterranean formation (304) has a plurality of geologicalstructures (306 a-306 d). As shown, the formation has a sandstone layer(306 a), a limestone layer (306 b), a shale layer (306 c), and a sandlayer (306 d). A fault line (307) extends through the formation. Thestatic data acquisition tools are preferably adapted to measure theformation and detect the characteristics of the geological structures ofthe formation.

While a specific subterranean formation (304) with specific geologicalstructures are depicted, it will be appreciated that the formation maycontain a variety of geological structures. Fluid may also be present invarious portions of the formation. Each of the measurement devices maybe used to measure properties of the formation and/or its underlyingstructures. While each acquisition tool is shown as being in specificlocations along the formation, it will be appreciated that one or moretypes of measurement may be taken at one or more location across one ormore oilfields or other locations for comparison and/or analysis.

The data collected from various sources, such as the data acquisitiontools of FIG. 3, may then be evaluated. Typically, seismic datadisplayed in the static data plot (308 a) from the data acquisition tool(302 a) is used by a geophysicist to determine characteristics of thesubterranean formation (304). Core data shown in static plot (308 b)and/or log data from the well log (308 c) is typically used by ageologist to determine various characteristics of the geologicalstructures of the subterranean formation (304). Production data from theproduction graph (308 d) is typically used by the reservoir engineer todetermine fluid flow reservoir characteristics.

FIG. 4 shows an oilfield (400) for performing production operations. Asshown, the oilfield has a plurality of wellsites (402) operativelyconnected to a central processing facility (454). The oilfieldconfiguration of FIG. 4 is not intended to limit the scope of theinvention. Part or all of the oilfield may be on land and/or sea. Also,while a single oilfield with a single processing facility and aplurality of wellsites is depicted, any combination of one or moreoilfields, one or more processing facilities and one or more wellsitesmay be present.

Each wellsite (402) has equipment that forms a wellbore (436) into theearth. The wellbores extend through subterranean formations (406)including reservoirs (404). These reservoirs (404) contain fluids, suchas hydrocarbons. The wellsites draw fluid from the reservoirs and passthem to the processing facilities via surface networks (444). Thesurface networks (444) have tubing and control mechanisms forcontrolling the flow of fluids from the wellsite to the processingfacility (454).

FIG. 5 shows a schematic view of a portion (or region) of the oilfield(400) of FIG. 4, depicting a producing wellsite (402) and surfacenetwork (444) in detail. The wellsite (402) of FIG. 5 has a wellbore(436) extending into the earth therebelow. As shown, the wellbores (436)has already been drilled, completed, and prepared for production fromreservoir (404).

Wellbore production equipment (564) extends from a wellhead (566) ofwellsite (402) and to the reservoir (404) to draw fluid to the surface.The wellsite (402) is operatively connected to the surface network (444)via a transport line (561). Fluid flows from the reservoir (404),through the wellbore (436), and onto the surface network (444). Thefluid then flows from the surface network (444) to the processfacilities (454).

As further shown in FIG. 5, sensors (S) are located about the oilfield(400) to monitor various parameters during oilfield operations. Thesensors (S) may measure, for example, pressure, temperature, flow rate,composition, and other parameters of the reservoir, wellbore, surfacenetwork, process facilities and/or other portions (or regions) of theoilfield operation. These sensors (S) are operatively connected to asurface unit (534) for collecting data therefrom. The surface unit maybe, for example, similar to the surface unit (134) of FIGS. 1A-D.

One or more surface units (534) may be located at the oilfield (400), orlinked remotely thereto. The surface unit (534) may be a single unit, ora complex network of units used to perform the necessary data managementfunctions throughout the oilfield (400). The surface unit may be amanual or automatic system. The surface unit may be operated and/oradjusted by a user. The surface unit is adapted to receive and storedata. The surface unit may also be equipped to communicate with variousoilfield equipment. The surface unit may then send command signals tothe oilfield in response to data received or modeling performed.

As shown in FIG. 5, the surface unit (534) has computer facilities, suchas memory (520), controller (522), processor (524), and display unit(526), for managing the data. The data is collected in memory (520), andprocessed by the processor (524) for analysis. Data may be collectedfrom the oilfield sensors (S) and/or by other sources. For example,oilfield data may be supplemented by historical data collected fromother operations, or user inputs.

The analyzed data (e.g., based on modeling performed) may then be usedto make decisions. A transceiver (not shown) may be provided to allowcommunications between the surface unit (534) and the oilfield (400).The controller (522) may be used to actuate mechanisms at the oilfield(400) via the transceiver and based on these decisions. In this manner,the oilfield (400) may be selectively adjusted based on the datacollected. These adjustments may be made automatically based on computerprotocol and/or manually by an operator. In some cases, well plans areadjusted to select optimum operating conditions or to avoid problems.

To facilitate the processing and analysis of data, simulators may beused to process the data for modeling various aspects of the oilfieldoperation. Specific simulators are often used in connection withspecific oilfield operations, such as reservoir or wellbore simulation.Data fed into the simulator(s) may be historical data, real time data orcombinations thereof. Simulation through one or more of the simulatorsmay be repeated or adjusted based on the data received.

As shown, the oilfield operation is provided with wellsite andnon-wellsite simulators. The wellsite simulators may include a reservoirsimulator (340), a wellbore simulator (342), and a surface networksimulator (344). The reservoir simulator (340) solves for hydrocarbonflow through the reservoir rock and into the wellbores. The wellboresimulator (342) and surface network simulator (344) solves forhydrocarbon flow through the wellbore and the surface network (444) ofpipelines. As shown, some of the simulators may be separate or combined,depending on the available systems.

The non-wellsite simulators may include process (346) and economics(348) simulators. The processing unit has a process simulator (346). Theprocess simulator (346) models the processing plant (e.g., the processfacilities (454)) where the hydrocarbon(s) is/are separated into itsconstituent components (e.g., methane, ethane, propane, etc.) andprepared for sales. The oilfield (400) is provided with an economicssimulator (348). The economics simulator (348) models the costs of partor the entire oilfield (400) throughout a portion or the entire durationof the oilfield operation. Various combinations of these and otheroilfield simulators may be provided.

While high quality petroleum reservoirs have been successfully exploredand exploited for producing oil and gas. Large reservoirs areincreasingly difficult to find and producing reservoirs have problemsthat need to be quickly diagnosed and remedied. Therefore, honoring allrelevant measurements to enable on-time decision making is necessary foroilfield operations. The oilfield operations generates a large amount ofpressure and production rate data (e.g., data generated from sensors (S)and/or data acquisition tools disposed throughout the oilfield asdescribed with respect to FIGS. 1A-D and 2-5 above), some of which canbe measured continuously in real-time. In addition, there are dataacquired sporadically, such as well logs and formation test data (e.g.,the well log (308 c) and seismic trace (308 d) of FIG. 3). Timely andmethodical interpretation of this data can provide insight into thestatus of the well and the reservoir as well as advanced notice topotentially detrimental events.

A workflow is a sequence of steps, organized into routines orsubroutines—some of which may be quite complex—that are carried out toachieve a particular goal. Each step receives input in various formats,ranging from digital files or spreadsheets to expert commentary. Thisinput is then processed using a predefined mode, such as a reservoirsimulator, spreadsheet analysis, or structured discussions and meetings.The resulting output is utilized in subsequent steps. The goal for mostoilfield asset teams is to arrive at an answer that will be used asinput for another process, or which will be used to drive a decision.Repetitive workflows can often be automated, freeing personnel to attendto non-routine tasks.

The present invention relates to simulating oilfield workflows using agridless analytical simulator. In one or more embodiments of theinvention, the computation efficiency of the gridless analyticalsimulator enables the integration of various sources of data atdifferent frequencies in one integrated application, which allows userto step from a single well evaluation & interpretation to multi-well,multi-phase, and/or multi-event diagnostic in a synchronized mode. Inone or more embodiments of the invention, oilfield workflows may besimulated by this fast gridless analytical simulator for handlingpressure transient data and performing interpretation of key performanceindicators during the well/field production life. In one or moreembodiments of the invention, these capabilities allow oilfieldworkflows to monitor and analyze data, anticipate and identify events,and to perform real-time diagnostics and interpretation during theentire life of producing wells.

In one or more embodiments of the invention, the gridless analyticalsimulator, described below, supports several well configurations andreservoir conditions including vertical, deviated, horizontal, andfractured wells, single and multiple layer heterogeneous reservoir,single phase and multi-phase flowing conditions, and is capable oftaking into account superposition effect in multi-well and multi ratescenarios. In one or more embodiments of the invention, specialreservoir condition, such as interference effects of multiple wells atdifferent events, may be simulated including surface constrains,pressure transient or rate transient events, etc.

In one or more embodiments of the invention, the gridless analyticalsimulator may be used either in automatic history matching mode or inprediction mode. The automatic history matching mode aims to compute inreal time, key reservoir and well parameters such as reservoir pressure,well skin, effective permeability and well productivity. Subsequently,the prediction mode predicts well and reservoir performance inreal-time. The prediction mode is a component to integrate more commonproduction engineer analysis that is used to manage a reservoir, such aswell test validation and back allocation correction and forecast in realtime, among others.

In one or more embodiments of the invention, the gridless analyticalsimulator may be used to integrate and keep alive the interaction of themultiple oilfield workflow sub-processes, such as data integration(sources, frequency, etc.), data preparation using techniques such aswavelets transforms to reduce data, remove noise & outliers andtransient identification, alarm management system to monitor and controlKPI, pressure transient interpretation, automatic model identificationusing neural networks and systems identification including the use ofdeconvolution, back allocation, rate reconstruction and well testvalidation, production (rate and pressure) forecast, reporting andvisualization, and/or other suitable oilfield workflow sub-processes.

FIGS. 6 and 7 show exemplary oilfield workflows modeled using thegridless analytical simulator. FIG. 6 is a flow chart of a permanentdownhole pressure gauge (PDG) workflow in an oilfield (e.g., theoilfield (300) of FIG. 3). One of the objectives of the PDG workflow isto enable a lifecycle process to maximize hydrocarbon producingperformance of the reservoir over its full life cycle. This is achievedby using a gridless analytical simulator (e.g., a version of thereservoir simulator (340) of FIG. 5), which is described in detail belowand can be configured to simulate an interference effect, for examplefrom multiple wellsites of the oilfield (300) in FIG. 3. In the PDGworkflow, real-time pressure data is obtained for the gridlessanalytical simulator from a permanent down-hole pressure gauge (e.g.,the data acquisition tool (302 d) of FIG. 3) (Step 601). The real-timepressure data is filtered, for example, by using a wavelet decompositiontechnique to remove outlier(s), noise, and identify transients (Step613). The transients may result from a changing oil production rate orshutting down and turning up the production. The identified transientsmay be used to mark a time interval for simulation sessions. The largeamount of real-time raw data may be sampled to reduce to the filtereddata to a manageable amount, while retaining all the relevantcharacteristics of the original larger data set.

The flow rate data may be obtained for the gridless analytical simulatorusing a variety of methods. In some examples, the flow rate data isobtained through real-time measurement (e.g., the fluid flow rate dataplot (308 d) of FIG. 3) using sensors (e.g., data acquisition tool (302d) of FIG. 3) disposed throughout the oilfield (Step 603). In someexamples, missing periods of the real-time measurement may exist, whichmay be supplemented with flow rate re-construction, for example based ontubing head or bottom hole pressure measurement (Step 604). Thereal-time flow rate data (if available) is also filtered in a similarfashion as filtering of the real-time pressure data (Step 605). In otherexamples, the real-time flow rate measurement may not be available (Step602). In such cases, the offline flow rate data is obtained, for exampleby a method of back allocation using total volume at the point of sales,well test data, and/or downtime measurement at a well (Step 606). Theoffline flow rate data may also be supplemented with flow ratere-construction, for example, based on tubing head or bottom holepressure measurement (Step 612).

A set of alarm conditions are calculated based on the real-time dataafter filtering (Step 607). The alarms may include, for example drawdownalarm, downtime alarm, etc. If the alarm is triggered, detaileddiagnostics are performed thereafter.

Within the gridless analytical simulator, many parameters may be used toconfigure an appropriate model for simulating the oilfield (e.g., theoilfield (300) of FIG. 3). In some examples, the model may be determinedmanually. The model may be identified by using a neural network methodbased on, for example, rate of change of the real-time pressure data(Step 608). The model may be further configured based on staticparameters obtained through geological surveys (e.g. as depicted in FIG.1 and FIG. 3 above).

Once the model is identified and the simulator is configured, real-timesimulation results are then generated (Step 609). The real-timesimulation may include a history matching of key parameters and aprediction of the production rates and reservoir pressure over time. Thehistory matching may be performed as a calibration step at the beginningof a simulation session marked by an identified transient from a changeof production rate and/or shutting down and turning up of theproduction. The real-time simulation results may be delivered in anautomatic workflow (i.e., the PDG workflow) with real-time plotting ofthe key parameters and alarm setting based on pre-determined criteria.The key parameters for the history matching and the real-time plottingmay include the reservoir pressure, well skin, effective permeability,and well productivity, etc. The model is automatically updated if thepredicted performance diverges from the actual performance by more thana pre-determined limit (Step 610).

In Step 611, the oilfield operation is performed based on the real-timesimulation results. For example, the real-time plotting in thesimulation results may be analyzed to determine a trend of a wellboreskin, and the oilfield operation performed includes scheduling aworkover operation to reduce the wellbore skin. In another example, thereal-time plotting in the simulation results may be analyzed todetermine a trend of effective permeability, and the oilfield operationperformed includes determining a re-completion strategy, such asscheduling an artificial lift operation.

FIG. 7 is a flow chart of a gas field workflow in a gas field, forexample, gas may be produced in the oilfield operations depicted inFIGS. 1A-1D and 2-5 above. Initially, the flow rate data is obtainedthrough real-time measurement (e.g., the fluid flow rate data plot (308d) of FIG. 3) using sensors (e.g., data acquisition tool (302 d) of FIG.3) disposed throughout the oilfield (Step 701). In some examples,missing periods of the real-time measurement may exist. These missingperiods may be supplemented with flow rate re-construction, for example,based on tubing head or bottom hole pressure measurement. The real-timeflow rate data is also filtered. The filtering functionality includes,for example de-noising using wavelets decomposition, outlier removal,transient identification, data reduction, etc.

As gas wells often may not be equipped with a permanent down holepressure gauge. A set of first level alarm conditions are calculatedbased on the real-time flow rate data and basic historic bottom hole ortubing head pressure measurements (Step 702). The alarms may include,for example, drawdown alarm, downtime alarm, etc. If the alarm istriggered, detailed diagnostics are performed thereafter.

Next, a determination is made as to whether real-time measurement isavailable for bottom hole or tubing head pressure (Step 703). If neitherbottom hole nor tubing head pressure measurements is available, offlinepressure data is obtained (if available), for example, using historicaldata and/or by spot measurement (Step 708). The processed real-time flowrate data, and the offline pressure data (if available) are then used tocompute key reservoir parameters such as total skin factor,permeability, drainage area, etc. using evaluation method withoutreal-time pressure data, for example, a non-linear regression model(Step 710).

If real-time pressure measurement is available (Step 703), thereliability of the analysis may increase by obtaining pressure data fromeither bottom hole or tubing head (Step 704). The real-time pressuredata obtained this way also involve a filtering step, which includesde-noising, outlier removal, transient identification, and sampling fordata reduction.

The reservoir model for a gridless analytical simulator is thenidentified (Step 705). The model may be identified by using a neuralnetwork method based on, for example, hydraulic flow units obtained frompre-processed logs containing information such as layer thickness,porosity, effective permeability, and saturation dependentpetro-physical properties. In this step, the model may be furtherconfigured based on a history matching method of these key parameters.

Once the model is identified and the simulator is configured, real-timesimulation results are then generated (Step 706). The real-timesimulation includes a history matching of key parameters and aprediction of the production rates and reservoir pressure over time. Thehistory matching may be performed as a calibration step at the beginningof a simulation session marked by an identified transient from a changeof production rate and/or a shutting down and a turning up of theproduction. The real-time simulation results can be delivered in anautomatic workflow (i.e., the gas field workflow) with real-timeplotting of the key parameters and alarm setting based on pre-determinedcriteria. The key parameters for the history matching and the real-timeplotting may include the reservoir pressure, well skin, effectivepermeability, and well productivity, etc. The model is automaticallyupdated if the predicted performance diverges from the actualperformance by more than a pre-determined limit (Step 707).

In Step 711, the oilfield operation is performed based on the real-timesimulation results.

FIG. 8 shows an exemplary schematic diagram of a reservoir modeled in agridless analytical simulator. In FIG. 8, the reservoir (800) (a portionof which may correspond to the reservoir (404) depicted in FIG. 4 andFIG. 5 above) is represented as a series of N vertically stacked cuboids(or layers) (801), where each of the N cuboids is indexed from 1 throughN by an index j. The reservoir (800) is bounded by the planes passingthrough x=0, x=a; y=0, y=b; z=0, z=d_(N) . Layer j has porosity φ_(j)and permeability k_(xj), k_(yj), k_(zj) in the x, y and z directionsrespectively. The scale of the reservoir (800) drawn in FIG. 8 may besubstantially larger than the scale used in FIG. 3, FIG. 4, and FIG. 5.

For example, portions of these cuboids (801) may correspond to thegeological structures (306 a-306 d) of FIG. 3. The reservoir (800) maybe penetrated by multiple wells such as vertical wells (802), horizontalwells (803), and deviated wells (804). The wells (802, 803, 804) may befractured or un-fractured, the fracture(s) may be naturally occurring orinduced by hydraulic fracturing process (not shown). The hydraulicfractures may have finite or infinite conductivity. The reservoirboundary may be modeled as no-flow, constant pressure, or a combinationthereof. Even though the wells (802, 803, 804) are represented as aline, suitable corrections may be applied in the model to account forwellbore storage effects and finite wellbore radius. Interference (orsuperposition) effects from multiple wells in the oilfield are accountedfor in the model.

In one or more embodiments of the invention, a gridless analyticalsimulator may be developed for the vertically stacked system of layersdescribed above. Specifically, an analytic solution within each layercan be derived using a method of integral transforms. In one or moreembodiments of the invention, the crossflow between layers are accountedfor by coupling these analytic solutions together and solving Fredholmintegral equations to obtain the flux field at the layer interfaces. Thetime evolution of these fluxes is governed by a Volterra integralequation. In one or more embodiments of the invention, the form of theseequations allows for stopping a model execution and then restarting fromthe exact terminated state.

In one or more embodiments of the invention, a general solution forhydrocarbon production can be formulated based on initial and boundaryconditions and the governing equations listed in TABLE 1.

TABLE 1 $\begin{matrix}\begin{matrix}{{\frac{\partial{p_{j}\left( {0,y,z,t} \right)}}{\partial x} = {{- \left( \frac{\mu}{k_{x}} \right)_{j}}{\psi_{0{{yz}j}}\left( {y,z,t} \right)}}},{\frac{\partial{p_{j}\left( {a,y,z,t} \right)}}{\partial x} = {{- \left( \frac{\mu}{k_{x}} \right)_{j}}{\psi_{ayzj}\left( {y,z,t} \right)}}},} \\{{\frac{\partial{p_{j}\left( {x,0,z,t} \right)}}{\partial y} = {{- \left( \frac{\mu}{k_{x}} \right)_{j}}{\psi_{x0zj}\left( {x,z,t} \right)}}},{\frac{\partial{p_{j}\left( {x,b,z,t} \right)}}{\partial y} = {{- \left( \frac{\mu}{k_{y}} \right)_{j}}{\psi_{xbzj}\left( {x,z,t} \right)}}},{d_{j} < z < d_{j + 1}},} \\{{{\forall j} = 0},1,\ldots,{{\aleph - {1.\mspace{14mu} {At}\mspace{14mu} z}} = d_{0}},{\frac{\partial{p\left( {x,y,d_{0},t} \right)}}{\partial z} = {{- \left( \frac{\mu}{k_{z}} \right)_{0}}{\psi_{{xy}\; 0\; 0}\left( {x,y,t} \right)}}},{{{and}\mspace{14mu} {at}\mspace{14mu} z} = d_{\aleph}},} \\{{\frac{\partial{p\left( {x,y,d_{\aleph},t} \right)}}{\partial z} = {{- \left( \frac{\mu}{k_{z}} \right)_{\aleph}}{\psi_{{xyd}\aleph}\left( {x,y,t} \right)}}},{0 < x < a},{{0 < y < {{b.\mspace{14mu} {At}}\mspace{14mu} {the}\mspace{14mu} {interface}\mspace{14mu} z}} = d_{j}},}\end{matrix} & (0.1)\end{matrix}$ $\; \begin{matrix}{{\psi_{j}\left( {x,z,t} \right)} = {{{- \left( \frac{k_{z}}{\mu} \right)_{j}}\left( \frac{\partial{p_{j}\left( {x,y,d_{j},t} \right)}}{\partial y} \right)} = {{- \left( \frac{k_{z}}{\mu} \right)_{j - 1}}\left( \frac{\partial{p_{j - 1}\left( {x,y,d_{j},t} \right)}}{\partial y} \right)\mspace{11mu} {and}}}} & \; \\{{{{\overset{\Cup}{\lambda}}_{j}{\psi_{j}\left( {x,z,t} \right)}} = \left\{ {{p_{{j\ldots}\; 1}\left( {x,y,d_{j},t} \right)} - {p_{j}\left( {x,y,d_{j},t} \right)}} \right\}},{{\forall j} = 1},\ldots \;,{\aleph - {1.\mspace{14mu} {The}\mspace{14mu} {initial}\mspace{14mu} {pressure}}}} & \; \\{{{p_{j}\left( {x,y,z,0} \right)} = {{{{\phi_{j}\left( {x,y,z} \right)}.\mspace{14mu} {In}}\mspace{14mu} {the}\mspace{14mu} {interval}\mspace{14mu} d_{j}} \leq z \leq d_{j + 1}}},{j = 0},1,\ldots \;,{\aleph - 1},{{we}\mspace{14mu} {find}\mspace{14mu} p_{j}},{{the}\mspace{14mu} {pressure}\mspace{14mu} {response}}} & \; \\{{{corresponding}\mspace{14mu} {any}\mspace{14mu} {perturbation}},{{from}\mspace{14mu} {the}\mspace{14mu} {partial}\mspace{14mu} {differential}\mspace{14mu} {equation}}} & \; \\{\frac{\partial p_{j}}{\partial t} = {{\eta_{xj}\frac{\partial^{2}p_{j}}{\partial x^{2}}} + {\eta_{yj}{\frac{\partial^{2}p_{j}}{\partial y^{2}}++}\eta_{zj}\frac{\partial^{2}p_{j}}{\partial z^{2}}} + {{U\left( {t - t_{0j}} \right)}\frac{q_{j}\left( {t - t_{0j}} \right)}{\left( {\varphi c}_{t} \right)_{j}}{\delta \left( {x - x_{0j}} \right)}{\delta \left( {y - y_{0j}} \right)}{\delta \left( {z - z_{0j}} \right)}}}} & \;\end{matrix}$

In the general solution, the hydrocarbon production occurs throughmultiple vertical or horizontal wells (e.g., vertical wells (802) andhorizontal wells (803)), multiple deviated wells (e.g., deviated wells(804)), and fractures.

The multiple vertical or horizontal wells are modeled as line sources offinite lengths [y_(02ij)−y_(01ij)], [z_(02ij)−z_(01ij)],[x_(02ij)−x_(01ij)] passing through:

(x_(0ij), y_(0ij)) for t=1, 2 . . . , L_(i)

(y_(0ij), z_(0ij)) for t=L_(i)+2 . . . , M_(i)

(x_(0ij), z_(0ij)) for t=M_(i)+1 . . . , N_(i)

The multiple deviated wells are modeled as [(z_(02ij)−z_(01ij)) sin∂θ_(0tj)], which passes through (x_(0ij), y_(0ij), z_(0ij)) fort=N_(t)+1, . . . , N_(d).

The fractures are modeled as rectangle sources of finite area[x_(02tj)−x_(01tj)][y_(02tj)−y_(01tj)],[y_(02tj)−y_(01tj)][z_(02tj)−z_(01tj)], and[x_(02tj)−x_(01tj)][z_(02tj)−z_(01tj)], which passes through:

z_(0tj) for t=N_(d)+1, . . . , L_(r)

x_(0tj) for t=L_(r)+1, . . . , M_(r)

y_(0tj) for t=M_(r)+1, . . . , N_(r)

(L_(t)<M_(t)<N_(t)<N_(d)<L_(r)<M_(r)<N_(r))

The pressure solution at any given point [x, y, z] in the reservoir attime t and the derivation to arrive at a set of general expressions isgiven as the equations (0.2) through (0.8) listed in TABLE 2 below.

TABLE 2 $\begin{matrix}\begin{matrix}{p_{j} = {\frac{1}{4\left( {\varphi c}_{t} \right)_{j}{ab}}{\sum\limits_{\iota = 1}^{L_{t}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {x - x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {x + x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {y - y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {y + y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {z - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z + z_{01{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{xj}\tau}} \right)} -} \right.}} \\{{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z - z_{02{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z + z_{02{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{xj}\tau}} \right)}} \right\} d\; \tau} +}\end{matrix} & (0.2)\end{matrix}$ $\begin{matrix}{{~~~~~~~}{{+ \frac{1}{4\left( {\varphi c}_{t} \right)_{j}{b\left( {d_{j + 1} - d_{j}} \right)}}}{\sum\limits_{\iota = {L_{l} + 1}}^{M_{l}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {y - y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {y + y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {z - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{xj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {z + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} -} \right.}} \\{{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{{02}{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} {d\tau}} +}\end{matrix}\quad$ $\begin{matrix}{{~\,~~~~~~}{{+ \frac{1}{4\left( {\varphi c}_{t} \right)_{j}{b\left( {d_{j + 1} - d_{j}} \right)}}}{\sum\limits_{\iota = {M_{l} + 1}}^{N_{l}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {y - y_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {x + x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {z - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {z + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} -} \right.}} \\{{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} {d\tau}} +}\end{matrix}\quad$ $\begin{matrix}{\mspace{40mu} {{+ \frac{1}{8\left( {\varphi c}_{t} \right)_{j}{{ab}\left( {d_{j + 1} - d_{j}} \right)}}} \times}} \\{{\times {\sum\limits_{\iota = {N_{l} + 1}}^{N_{d}}{{U\left( {t - t_{0{\iota j}}} \right)}\sin \mspace{11mu} \vartheta_{0{\iota j}}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)}{\int_{z\; 01{\iota j}}^{z_{02{\iota j}}}\left\lbrack \left\{ {{\Theta_{3}\left( {{\frac{\pi}{2a}\left\{ {x - {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\cot \; \vartheta_{0{\iota j}}\cos \; \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{a})}^{2}}{\eta {xj}\tau}}} \right)} +} \right. \right.}}}}}}} \\{\left. {+ {\Theta_{3}\left( {{\frac{\pi}{2a}\left\{ {x + {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\cot \; \vartheta_{0{\iota j}}\cos \; \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{a})}^{2}}{\eta {xj}\tau}}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}\left( {{\frac{\pi}{2b}\left\{ {y - {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\cot \; \vartheta_{0{\iota j}}\cos \; \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{b})}^{2}}{\eta {yj}\tau}}} \right)} +} \right.}} \\{\left. {+ {\Theta_{3}\left( {{\frac{\pi}{2b}\left\{ {y + {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\cot \; \vartheta_{0{\iota j}}\cos \; \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{b})}^{2}}{\eta {yj}\tau}}} \right)}} \right\} \times} \\{{\left. {\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {z - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta {yj}\tau}}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {z + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta {yj}\tau}}} \right)}} \right\}} \right\rbrack {dz}_{0{\iota j}}d\; \tau} +}\end{matrix}\quad$ $\begin{matrix}{\mspace{40mu} {{+ \frac{1}{2\left( {\varphi c}_{t} \right)_{j}\left( {d_{j + 1} - d_{j}} \right)}}{\sum\limits_{\iota = {N_{d} + 1}}^{L_{r}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} -} \right.}} \\{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} -} \right.}} \\{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times} \\{{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {z - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{xj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {z + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{xj}\tau}} \right)}} \right\} d\; \tau} +}}\end{matrix}\quad$ $\begin{matrix}{\mspace{40mu} {{+ \frac{1}{2\left( {\varphi c}_{t} \right)_{j}a}}{\sum\limits_{\iota = {L_{r} + 1}}^{M_{r}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {x - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {x + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} -} \right.}} \\{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {z - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z + z_{01{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} -} \right.}} \\{{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z - z_{02{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z + z_{02{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} d\; \tau} +}\end{matrix}\quad$ $\begin{matrix}{\mspace{40mu} {{+ \frac{1}{2\left( {\varphi c}_{t} \right)_{j}b}}{\sum\limits_{\iota = {M_{r} + 1}}^{N_{r}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} -} \right.}} \\{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {y - y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} - {{\Theta_{3}\left( {\frac{\pi \left( {y + y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} \times}} \right.}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {z - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z + z_{01{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} -} \right.}} \\{{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z - z_{02{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z + z_{02{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} d\; \tau} +}\end{matrix}\quad$ $\begin{matrix}{\mspace{40mu} {{+ \frac{1}{4\left( {\varphi c}_{t} \right)_{j}{{ab}\left( {d_{j + 1} - d_{j}} \right)}}}{\int_{0}^{t}{\int_{0}^{a}{\int_{0}^{b}\left\lbrack {{{\psi_{j}\left( {u,v,\tau} \right)}\Theta_{3}\left\{ {\frac{\pi \left( {z - d_{j}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} -} \right.}}}}} \\{\left. {{- {\psi_{j + 1}\left( {u,v,\tau} \right)}}\Theta_{4}\left\{ {\frac{\pi \left( {z - d_{j}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} \right\rbrack \times} \\{{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {x - u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {x + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}}} \right\rbrack \times}} \\{{{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {y - u} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {y + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}}} \right\rbrack {dudvdr}} +}}\end{matrix}\quad$ $\begin{matrix}{\mspace{40mu} {{+ \frac{1}{4\left( {\varphi c}_{t} \right)_{j}{{ab}\left( {d_{j + 1} - d_{j}} \right)}}} \times}} \\{{\times {\int_{0}^{t}{\int_{0}^{b}{\int_{0}^{d_{j + 1} - d_{j}}{\left\lbrack {{{\psi_{0{yzj}}\left( {v,w,\tau} \right)}\Theta_{3}\left\{ {\frac{\pi x}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right)} - {{\psi_{ayzj}\left( {v,w,\tau} \right)}\Theta_{4}\left\{ {\frac{\pi x}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right)}} \right\} \times}}}}}} \\{{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {y - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {y + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}}} \right\rbrack \times}} \\{{{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {z - d_{j} - w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {z - d_{j} + w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}}} \right\rbrack {dvdwd}\; \tau} +}}\end{matrix}\quad$ $\begin{matrix}{\mspace{50mu} {{+ \frac{1}{4\left( {\varphi c}_{t} \right)_{j}{{ab}\left( {d_{j + 1} - d_{j}} \right)}}} \times}} \\{{\times {\int_{0}^{t}{\int_{0}^{a}{\int_{0}^{d_{j + 1} - d_{j}}{\left\lbrack {{{\psi_{x0{zj}}\left( {u,w,\tau} \right)}\Theta_{3}\left\{ {\frac{\pi y}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right)} - {{\psi_{xbzj}\left( {u,w,\tau} \right)}\Theta_{4}\left\{ {\frac{\pi y}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right)}} \right\} \times}}}}}} \\{{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {x - u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {x + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}}} \right\rbrack \times}} \\{{{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {z - d_{j} - w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {z - d_{j} + w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}}} \right\rbrack {dudwd}\; \tau} +}}\end{matrix}\quad$ $\begin{matrix}{\mspace{50mu} {{+ \frac{1}{8{{ab}\left( {d_{j + 1} - d_{j}} \right)}}} \times}} \\{{\times {\int_{0}^{d_{j + 1} - d_{j}}{\int_{0}^{b}{\int_{0}^{a}{{\phi_{j}\left( {u,v,{w\; + d_{j}}} \right)}\left\lbrack {\left\{ {{\Theta_{3}\left\{ {\frac{\pi\left( {x - u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}t}} \right)} + {\Theta_{3}\left\{ {\frac{\pi \left( {x + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}t}} \right)}} \right\} \times} \right.}}}}}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {y - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}t}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {y + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}t}} \right)}} \right\} \times}} \\{\left. {\times \left\{ {{\Theta_{3}\left\{ {\frac{\pi \left( {z - d_{j} - w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}t}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {z - d_{j} + w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}t}} \right\}}} \right\}} \right\rbrack {dudvdw}}\end{matrix}{\quad {{{{where}\mspace{14mu} \eta_{xj}} = \left( \frac{k_{z}}{\varphi \; c_{t}\mu} \right)_{j}},{\eta_{yj} = {{\left( \frac{k_{y}}{\varphi \; c_{t}\mu} \right)_{j}\mspace{14mu} {and}\mspace{14mu} \eta_{zj}} = {{\left( \frac{k_{z}}{\varphi \; c_{t}\mu} \right)_{j}.\mspace{14mu} {We}}\mspace{14mu} {employ}}}},\; {{in}\mspace{14mu} {time}\mspace{14mu} {domain}},{{the}\mspace{14mu} {interfacial}{boundary}\mspace{14mu} {{condition}.\mspace{14mu} {Substituting}}\mspace{14mu} {for}\mspace{14mu} {p_{j}\left( {x,y,d_{j},t} \right)}\mspace{14mu} {and}\mspace{14mu} {p_{j - 1}\left( {x,y,d_{j},t} \right)}\mspace{14mu} {from}\mspace{14mu} {equation}\mspace{14mu} (0.2)\mspace{14mu} {in}}}}$$\begin{matrix}{{{{{\overset{\Cup}{\lambda}}_{j}{\psi_{j}\left( {x,y,t} \right)}} = \left\{ {{p_{j - 1}\left( {x,y,d_{j},t} \right)} - {p_{j}\left( {x,y,d_{j},t} \right)}} \right\}},\mspace{31mu} {{\forall j} = 1},2,\ldots \;,{\aleph - 1}}{{we}\mspace{14mu} {obtain}\mspace{14mu} a\mspace{14mu} {three}\text{-}{point}\mspace{14mu} \text{recurrence}\mspace{14mu} {integral}\mspace{14mu} {equation}\mspace{14mu} {relationship}\mspace{14mu} {in}\mspace{14mu} {time}\mspace{14mu} {and}\mspace{14mu} {space}}} & \; \\\begin{matrix}{{{\overset{\Cup}{\lambda}}_{j}{\psi_{j}\left( {x,y,t} \right)}} = {{\int_{0}^{t}{\int_{0}^{a}{\int_{0}^{b}{{A_{j}\left( {x,u,{y.v},{t - \tau}} \right)}{\psi_{j}\left( {u,v,\tau} \right)}{dvdud}\; \tau}}}} +}} \\{{{+ {\int_{0}^{t}{\int_{0}^{a}{\int_{0}^{b}{{B_{j}\left( {x,u,y,v,{t - \tau}} \right)}{\psi_{j + 1}\left( {u,v,\tau} \right)}{dvdud}\; \tau}}}}} +}} \\{{{+ {\int_{0}^{t}{\int_{0}^{a}{\int_{0}^{b}{{C_{j}\left( {x,u,y,v,{t - \tau}} \right)}{\psi_{j - 1}\left( {u,v,\tau} \right)}{dvdud}\; \tau}}}}} + {\Omega_{j}\left( {x,y,t} \right)}}}\end{matrix} & (0.3) \\{{{{The}\mspace{14mu} {coefficients}\mspace{14mu} {of}\mspace{14mu} {the}\mspace{14mu} {recurrence}\mspace{14mu} {integral}\mspace{14mu} {equation}\mspace{14mu} (0.3)\mspace{14mu} {for}\mspace{14mu} d_{j}} < z < d_{j + 1}},{\forall_{j}{= 1}},2,\ldots \;,{\aleph - 1},{{are}\mspace{14mu} {given}\mspace{14mu} {by}}} & \; \\{{A_{j}\left( {x,u,y,v,t} \right)} = {{g_{j}\left( {x,u,y,v,t} \right)} + {g_{j - 1}\left( {x,u,y,v,t} \right)}}} & (0.4)\end{matrix}$ where $\begin{matrix}\begin{matrix}{{g_{j}\left( {x,u,y,v,t} \right)} = {{- \frac{\Theta_{3}\left\{ {0,e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}({t - \tau})}}} \right\}}{4\left( {\varphi c}_{t} \right)_{j}{{ab}\left( {d_{j + 1} - d_{j}} \right)}}} \times}} \\{{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {x - u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {x + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}}} \right\rbrack \times}} \\{{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {y - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {y + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}}} \right\rbrack}}\end{matrix} & (0.5) \\\begin{matrix}{{B_{j}\left( {x,u,y,v,t} \right)} = {{- \frac{\Theta_{4}\left\{ {0,e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{xj}({t - \tau})}}} \right\}}{4\left( {\varphi c}_{t} \right)_{j}{{ab}\left( {d_{j + 1} - d_{j}} \right)}}} \times}} \\{{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {x - u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {x + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}}} \right\rbrack \times}} \\{{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {y - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {y + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}}} \right\rbrack}}\end{matrix} & (0.6)\end{matrix}$ C_(j)(x, u, y, v, t) = B_(j−1)(x, u, y, v, t) (0.7) and$\begin{matrix}\begin{matrix}{{\Omega_{j}\left( {x,y,t} \right)} = {\frac{1}{4\left( {\varphi c}_{t} \right)_{j - 1}{ab}}{\sum\limits_{\iota = 1}^{L_{t}}{{U\left( {t - t_{{0{\iota j}} - 1}} \right)}{\int_{0}^{t - t_{{0{\iota j}} - 1}}{{q_{{\iota j} - 1}\left( {t - t_{{0{\iota j}} - 1} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {x - x_{{0{\iota j}} - 1}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{{zj} - 1}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {x + x_{{0{\iota j}} - 1}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{{zj} - 1}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {y - y_{{0{\iota j}} - 1}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{{yj} - 1}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {y + y_{{0{\iota j}} - 1}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{{yj} - 1}\tau}} \right)}} \right\} \times}}\end{matrix} & (0.8) \\\begin{matrix}{{~~~~~~~~~~~~~~~~~~~~~~~~~~~~}{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {d_{j} - z_{{01{\iota j}} - 1}} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j} - d_{j - 1}})}^{2}}\eta_{{zj} - 1}\tau}} \right)} -} \right.}} \\{{{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {d_{j} + z_{{01{\iota j}} - 1} - {2d_{j - 1}}} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j} - d_{j - 1}})}^{2}}\eta_{{zj} - 1}\tau}} \right)}} -}} \\{{{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {d_{j} - z_{{02{\iota j}} - 1}} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j} - d_{j - 1}})}^{2}}\eta_{{zj} - 1}\tau}} \right)}} +}} \\{{\left. {+ {\Theta_{3}^{\int}\left( {\frac{\pi \left( {d_{j} + z_{{02{\iota j}} - 1} - {2d_{j - 1}}} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j} - d_{j - 1}})}^{2}}\eta_{{zj} - 1}\tau}} \right)}} \right\} d\; \tau} +}\end{matrix} & \;\end{matrix}$ $\begin{matrix}{\mspace{110mu} {{+ \frac{1}{4\left( {\varphi c}_{t} \right)_{j - 1}{b\left( {d_{j} - d_{j - 1}} \right)}}}{\sum\limits_{\iota = {L_{t} + 1}}^{M_{t}}{{U\left( {t - t_{{0{\iota j}} - 1}} \right)}{\int_{0}^{t - t_{{0{\iota j}} - 1}}{{q_{{\iota j} - 1}\left( {t - t_{{0{\iota j}} - 1} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {y - y_{{0{\iota j}} - 1}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{{yj} - 1}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {y + y_{{0{\iota j}} - 1}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{{yj} - 1}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {d_{j} - z_{{0{\iota j}} - 1}} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j} - d_{j - 1}})}^{2}}\eta_{{zj} - 1}\tau}} \right)} +} \right.}} \\{\left. {+ {\Theta_{3}\left( {\frac{\pi \left( {d_{j} + z_{{0{\iota j}} - 1} - {2d_{j - 1}}} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j} - d_{j - 1}})}^{2}}\eta_{{zj} - 1}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{{01{\iota j}} - 1}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{{zj} - 1}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{{01{\iota j}} - 1}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{{zj} - 1}\tau}} \right)} -} \right.}} \\{{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{{02{\iota j}} - 1}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{{zj} - 1}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{{02{\iota j}} - 1}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{{zj} - 1}\tau}} \right)}} \right\} d\; \tau} +}\end{matrix}\quad$ $\begin{matrix}{{~~~~~~~~~~~~~~~~~~~~~~}{{+ \frac{1}{4\left( {\varphi c}_{t} \right)_{j - 1}{a\left( {d_{j} - d_{j - 1}} \right)}}}{\sum\limits_{\iota = {M_{t} + 1}}^{N_{t}}{{U\left( {t - t_{{0{\iota j}} - 1}} \right)}{\int_{0}^{t - t_{{0{\iota j}} - 1}}{{q_{{\iota j} - 1}\left( {t - t_{{0{\iota j}} - 1} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {x - x_{{0{\iota j}} - 1}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{{zj} - 1}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {x + x_{{0{\iota j}} - 1}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{{zj} - 1}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {d_{j} - z_{{0{\iota j}} - 1}} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j} - d_{j - 1}})}^{2}}\eta_{{zj} - 1}\tau}} \right)} +} \right.}} \\{\left. {+ {\Theta_{3}\left( {\frac{\pi \left( {d_{j} + z_{{0{\iota j}} - 1} - {2d_{j - 1}}} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j} - d_{j - 1}})}^{2}}\eta_{{zj} - 1}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{{01{\iota j}} - 1}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{{yj} - 1}\tau}} \right)} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{{01{\iota j}} - 1}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{{yj} - 1}\tau}} \right)} -} \right.}} \\{{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{{02{\iota j}} - 1}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{{yj} - 1}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{{02{\iota j}} - 1}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{{yj} - 1}\tau}} \right)}} \right\} d\; \tau} +}\end{matrix}\quad$ $\begin{matrix}{\mspace{110mu} {{+ \frac{1}{8\left( {\varphi c}_{t} \right)_{j - 1}{{ab}\left( {d_{j} - d_{j - 1}} \right)}}}{\sum\limits_{\iota = {N_{t} + 1}}^{N_{d}}{{U\left( {t - t_{{0{\iota j}} - 1}} \right)}\sin \mspace{11mu} \vartheta_{{0{\iota j}} - 1}{\int_{0}^{t - t_{{0{\iota j}} - 1}}{{q_{{\iota j} - 1}\left( {t - t_{{0{\iota j}} - 1} - \tau} \right)} \times}}}}}} \\{{\times {\int_{z_{{01{\iota j}} - 1}}^{z_{{02{\iota j}} - 1}}\left\lbrack \left\{ {{\Theta_{3}\left( {{\frac{\pi}{2a}\; \left\{ {x - {\left( {z_{{0{\iota j}} - 1} - \gamma_{{0{\iota j}} - 1}} \right)\cot \mspace{14mu} \vartheta_{{0{\iota j}} - 1}\mspace{11mu} \cos \mspace{11mu} \theta_{{0{\iota j}} - 1}}} \right\}},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{{xj} - 1}\tau}} \right)} +} \right. \right.}}} \\{\left. {+ {\Theta_{3}\left( {{\frac{\pi}{2a}\; \left\{ {x + {\left( {z_{{0{\iota j}} - 1} - \gamma_{{0{\iota j}} - 1}} \right)\cot \mspace{14mu} \vartheta_{{0{\iota j}} - 1}\mspace{11mu} \cos \mspace{11mu} \theta_{{0{\iota j}} - 1}}} \right\}},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{{zj} - 1}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}\left( {{\frac{\pi}{2b}\; \left\{ {y - {\left( {z_{{0{\iota j}} - 1} - \gamma_{{0{\iota j}} - 1}} \right)\cot \mspace{14mu} \vartheta_{{0{\iota j}} - 1}\mspace{11mu} \sin \mspace{11mu} \theta_{{0{\iota j}} - 1}}} \right\}},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{{yj} - 1}\tau}} \right)} +} \right.}} \\{\left. {+ {\Theta_{3}\left( {{\frac{\pi}{2b}\; \left\{ {y + {\left( {z_{{0{\iota j}} - 1} - \gamma_{{0{\iota j}} - 1}} \right)\cot \mspace{14mu} \vartheta_{{0{\iota j}} - 1}\mspace{11mu} \sin \mspace{11mu} \theta_{{0{\iota j}} - 1}}} \right\}},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{{yj} - 1}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {d_{j} - z_{{0{\iota j}} - 1}} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j} - d_{j - 1}})}^{2}}\eta_{{yj} - 1}\tau}} \right)} +} \right.}} \\{{\left. \left. {+ {\Theta_{3}\left( {\frac{\pi \left( {d_{j} + z_{{0{\iota j}} - 1} - {2d_{j - 1}}} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j} - d_{j - 1}})}^{2}}\eta_{{yj} - 1}\tau}} \right)}} \right\} \right\rbrack \; {dz}_{{0{\iota j}} - 1}d\; \tau} +}\end{matrix}\quad$ $\begin{matrix}{\mspace{104mu} {{+ \frac{1}{2\left( {\varphi c}_{t} \right)_{j - 1}\left( {d_{j} - d_{j - 1}} \right)}}{\sum\limits_{\iota = {N_{d} + 1}}^{L_{r}}{{U\left( {t - t_{{0{\iota j}} - 1}} \right)}{\int_{0}^{t - t_{{0{\iota j}} - 1}}{{q_{{\iota j} - 1}\left( {t - t_{{0{\iota j}} - 1} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{{01{\iota j}} - 1}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{{zj} - 1}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{{01{\iota j}} - 1}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{{zj} - 1}\tau}} \right)} -} \right.}} \\{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{{02{\iota j}} - 1}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{{zj} - 1}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{{02{\iota j}} - 1}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{{zj} - 1}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{{0{\iota j}} - 1}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{{yj} - 1}\tau}} \right)} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{{01{\iota j}} - 1}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{{yj} - 1}\tau}} \right)} -} \right.}} \\{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{{02{\iota j}} - 1}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{{yj} - 1}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{{02{\iota j}} - 1}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{{yj} - 1}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {d_{j} - z_{{0{\iota j}} - 1}} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j} - d_{j - 1}})}^{2}}\eta_{{zj} - 1}\tau}} \right)} +} \right.}} \\{{\left. {{+ \Theta_{3}}\left\{ {\frac{\pi \left( {d_{j} + z_{{0{\iota j}} - 1} - {2d_{j - 1}}} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j} - d_{j - 1}})}^{2}}\eta_{{zj} - 1}\tau}} \right)} \right\} d\; \tau} +}\end{matrix}\quad$ $\begin{matrix}{\mspace{110mu} {{+ \frac{1}{2\left( {\varphi c}_{t} \right)_{j - 1}a}}{\sum\limits_{\iota = {L_{r} + 1}}^{M_{r}}{{U\left( {t - t_{{0{\iota j}} - 1}} \right)}{\int_{0}^{t - t_{{0{\iota j}} - 1}}{{q_{{\iota j} - 1}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {x - x_{{01{\iota j}} - 1}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{{zj} - 1}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {x + x_{{01{\iota j}} - 1}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{{xj} - 1}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{{01{\iota j}} - 1}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{{yj} - 1}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{{01{\iota j}} - 1}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{{yj} - 1}\tau}} \right)} -} \right.}} \\{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{{02{\iota j}} - 1}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{{yj} - 1}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{{02{\iota j}} - 1}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{{yj} - 1}\tau}} \right)}} \right\} \times}\end{matrix}{\quad {\begin{matrix}{{~~~~~~~~~~~~~~~~~~~~~~~~~}{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {d_{j} - z_{{01{\iota j}} - 1}} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j} - d_{j - 1}})}^{2}}\eta_{{zj} - 1}\tau}} \right)} -} \right.}} \\{{{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {d_{j} + z_{{01{\iota j}} - 1} - {2d_{j - 1}}} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j} - d_{j - 1}})}^{2}}\eta_{{zj} - 1}\tau}} \right)}} -}} \\{{{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {d_{j} - z_{{02{\iota j}} - 1}} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j} - d_{j - 1}})}^{2}}\eta_{{zj} - 1}\tau}} \right)}} +}} \\{{\left. {+ {\Theta_{3}^{\int}\left( {\frac{\pi \left( {d_{j} + z_{{02{\iota j}} - 1} - {2d_{j - 1}}} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j} - d_{j - 1}})}^{2}}\eta_{{zj} - 1}\tau}} \right)}} \right\} \mspace{11mu} d\; \tau} +}\end{matrix}\quad}}$ $\begin{matrix}{\mspace{110mu} {{+ \frac{1}{2\left( {\varphi c}_{t} \right)_{j - 1}b}}{\sum\limits_{\iota = {M_{r} + 1}}^{N_{r}}{{U\left( {t - t_{{0{\iota j}} - 1}} \right)}{\int_{0}^{t - t_{{0{\iota j}} - 1}}{{q_{{\iota j} - 1}\left( {t - t_{{0{\iota j}} - 1} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{{01{\iota j}} - 1}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{{zj} - 1}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{{01{\iota j}} - 1}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{{zj} - 1}\tau}} \right)} -} \right.}} \\{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{{02{\iota j}} - 1}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{{zj} - 1}\tau}} \right)}} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{{02{\iota j}} - 1}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{{zj} - 1}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {y - y_{{0{\iota j}} - 1}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{{yj} - 1}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {y + y_{{0{\iota j}} - 1}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{{yj} - 1}\tau}} \right)}} \right\} \times}}\end{matrix}{\quad {\begin{matrix}{{~~~~~~~~~~~~~~~~~~~~~~~~~}{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {d_{j} - z_{{01{\iota j}} - 1}} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j} - d_{j - 1}})}^{2}}\eta_{{zj} - 1}\tau}} \right)} -} \right.}} \\{{{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {d_{j} + z_{{01{\iota j}} - 1} - {2d_{j - 1}}} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j} - d_{j - 1}})}^{2}}\eta_{{xj} - 1}\tau}} \right)}} -}} \\{{{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {d_{j} - z_{{02{\iota j}} - 1}} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j} - d_{j - 1}})}^{2}}\eta_{{zj} - 1}\tau}} \right)}} +}} \\{{\left. {+ {\Theta_{3}^{\int}\left( {\frac{\pi \left( {d_{j} + z_{{02{\iota j}} - 1} - {2d_{j - 1}}} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j} - d_{j - 1}})}^{2}}\eta_{{zj} - 1}\tau}} \right)}} \right\} \mspace{11mu} d\; \tau} +}\end{matrix}\quad}}$ $\begin{matrix}{\mspace{110mu} {{+ \frac{1}{4\left( {\varphi c}_{t} \right)_{j - 1}{{ab}\left( {d_{j} - d_{j - 1}} \right)}}}{\int_{0}^{t}{\int_{0}^{b}{\int_{0}^{({d_{j} - d_{j - 1}})}\left\lbrack {{{\psi_{{0{yzj}} - 1}\left( {v,w,\tau} \right)}{\Theta_{3}\left( {\frac{\pi x}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{{zj} - 1}{({t - \tau})}}}} \right)}} -} \right.}}}}} \\{\left. {{- {\psi_{{ayzj} - 1}\left( {v,w,\tau} \right)}}\Theta_{4}\left\{ {\frac{\pi x}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{{xj} - 1}{({t - \tau})}}}} \right)} \right\rbrack \times} \\{{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {y - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{{yj} - 1}{({t - \tau})}}}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {y + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{{yj} - 1}{({t - \tau})}}}} \right\}}} \right\rbrack \times}} \\{{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {d_{j} - d_{j - 1} - w} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j} - d_{j - 1}})}^{2}}{\eta_{{zj} - 1}{({t - \tau})}}}} \right\}} +} \right.}} \\{{\left. {~~}{{+ \Theta_{3}}\left\{ {\frac{\pi \left( {d_{j} - d_{j - 1} + w} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j} - d_{j - 1}})}^{2}}{\eta_{{zj} - 1}{({t - \tau})}}}} \right\}} \right\rbrack {dvdwd}\; \tau} +}\end{matrix}\quad$ $\begin{matrix}{\mspace{110mu} {{+ \frac{1}{4\left( {\varphi c}_{t} \right)_{j - 1}{{ab}\left( {d_{j} - d_{j - 1}} \right)}}}{\int_{0}^{t}{\int_{0}^{a}{\int_{0}^{d_{j} - d_{j - 1}}\left\lbrack {{{\psi_{{x0{zj}} - 1}\left( {u,w,\tau} \right)}\Theta_{3}\left\{ {\frac{\pi y}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{{yj} - 1}{({t - \tau})}}}} \right)} -} \right.}}}}} \\{\left. {{- {\psi_{{xbzj} - 1}\left( {u,w,\tau} \right)}}\Theta_{4}\left\{ {\frac{\pi y}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{{yj} - 1}{({t - \tau})}}}} \right)} \right\rbrack \times} \\{{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {x - u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{{zj} - 1}{({t - \tau})}}}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {x + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{{zj} - 1}{({t - \tau})}}}} \right\}}} \right\rbrack \times}} \\{{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {d_{j} - d_{j - 1} - w} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j} - d_{j - 1}})}^{2}}{\eta_{{zj} - 1}{({t - \tau})}}}} \right\}} +} \right.}} \\{{\left. {~~}{{+ \Theta_{3}}\left\{ {\frac{\pi \left( {d_{j} - d_{j - 1} + w} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j} - d_{j - 1}})}^{2}}{\eta_{{zj} - 1}{({t - \tau})}}}} \right\}} \right\rbrack {dudwd}\; \tau} +}\end{matrix}\quad$ $\begin{matrix}{\mspace{104mu} {{+ \frac{1}{8{{ab}\left( {d_{j} - d_{j - 1}} \right)}}} \times}} \\{{\times {\int_{0}^{d_{j} - d_{j - 1}}{\int_{0}^{b}{\int_{0}^{a}{{{\phi \left( {u,v,{w + d_{j - 1}}} \right)}\left\lbrack {{\Theta_{3}\left( {\frac{\pi\left( {x - u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{{zj} - 1}t}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {x + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{{zj} - 1}t}} \right)}} \right\rbrack} \times}}}}}} \\{{\times \left\lbrack {{\Theta_{3}\left( {\frac{\pi \left( {y - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{{yj} - 1}t}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {y + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{{yj} - 1}t}} \right)}} \right\rbrack \times}} \\{{\times \left\lbrack {{\Theta_{3}\left( {\frac{\pi \left( {d_{j} - d_{j - 1} - w} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j} - d_{j - 1}})}^{2}}\eta_{{zj} - 1}t}} \right)} +} \right.}} \\{{\left. {+ {\Theta_{3}\left( {\frac{\pi \left( {d_{j} - d_{j - 1} + w} \right)}{2\left( {d_{j} - d_{j - 1}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{{zj} - 1}t}} \right)}} \right\rbrack {dudvdw}} -}\end{matrix}\quad$ $\begin{matrix}{\mspace{115mu} {{- \frac{1}{2\left( {\varphi c}_{t} \right)_{j}{ab}}}{\sum\limits_{\iota = 1}^{L_{t}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {x - x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {x + x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {y - y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {y + y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times}} \\{{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {d_{j} - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {d_{j} - z_{02{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} d\; \tau} -}}\end{matrix}$ $\begin{matrix}{\mspace{110mu} {{- \frac{1}{2\left( {\varphi c}_{t} \right)_{j}{b\left( {d_{j + 1} - d_{j}} \right)}}} \times}} \\{{\times {\sum\limits_{\iota = {L_{t} + 1}}^{M_{t}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)}{\Theta_{3}\left( {\frac{\pi \left( {d_{j} - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {y - y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {y + y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{zj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} -} \right.}} \\{{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} d\; \tau} -}\end{matrix}\quad$ $\mspace{110mu} {\begin{matrix}{{- \frac{1}{2\left( {\varphi c}_{t} \right)_{j}{a\left( {d_{j + 1} - d_{j}} \right)}}} \times} \\{{\times {\sum\limits_{\iota = {M_{r} + 1}}^{N_{t}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)}{\Theta_{3}\left( {\frac{\pi \left( {d_{j} - d_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{xj}\tau}} \right)} \times}}}}}}\end{matrix}{\quad {\begin{matrix}{{~~~~~~~~~~~~~~~~~~~~~~~~~~}{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {x - x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{zj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {x + x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} -} \right.}} \\{{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} d\; \tau} -}\end{matrix}\quad}}}$ $\begin{matrix}{\mspace{115mu} {{- \frac{1}{4\left( {\varphi c}_{t} \right)_{j}{{ab}\left( {d_{j + 1} - d_{j}} \right)}}}{\sum\limits_{\iota = {N_{t} + 1}}^{N_{d}}{{U\left( {t - t_{0{\iota j}}} \right)}\sin \mspace{11mu} \vartheta_{0{\iota j}}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times {\int_{z_{01{\iota j}}}^{z_{02{\iota j}}}\left\lbrack \left\{ {{\Theta_{3}\left\{ {{\frac{\pi}{2a}\; \left\{ {x - {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\cot \mspace{14mu} \vartheta_{0{\iota j}}\mspace{11mu} \cos \mspace{11mu} \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} +} \right. \right.}}} \\{{{+ {\Theta_{3}\left( \left\{ {{\frac{\pi}{2a}\; \left\{ {x - {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\cot \mspace{14mu} \vartheta_{0{\iota j}}\mspace{11mu} \cos \mspace{11mu} \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{zj}\tau}} \right) \right\}}} \times}}\end{matrix}\quad$ $\begin{matrix}{{~~~~~~~~~~~~~~~~~~~~~~~~~~~}{\times \left\{ {{\Theta_{3}\left( {{\frac{\pi}{2b}\; \left\{ {y - {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\cot \mspace{14mu} \vartheta_{0{\iota j}}\mspace{11mu} \cos \mspace{11mu} \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +} \right.}} \\{\left. {+ {\Theta_{3}\left( {{\frac{\pi}{2b}\; \left\{ {y + {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\cot \mspace{14mu} \vartheta_{0{\iota j}}\mspace{11mu} \sin \mspace{11mu} \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times} \\{{\left. {\times {\Theta_{3}\left( {\frac{\pi \left( {d_{j} - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{yj}\tau}} \right)}} \right\rbrack {dz}_{0{\iota j}}d\; \tau} -}\end{matrix}\quad$ $\begin{matrix}{\mspace{110mu} {{- \frac{1}{\left( {\varphi c}_{t} \right)_{j}\left( {d_{j + 1} - d_{j}} \right)}}{\sum\limits_{\iota = {M_{d} + 1}}^{L_{r}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)}\Theta_{3}\left\{ {\frac{\pi \left( {d_{j} - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right) \times}}}}}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{zj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} -} \right.}} \\{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} -} \right.}} \\{{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} d\; \tau} -}\end{matrix}\quad$ $\begin{matrix}{\mspace{104mu} {{- \frac{1}{\left( {\varphi c}_{t} \right)_{j}a}}{\sum\limits_{\iota = {L_{r} + 1}}^{M_{r}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {x - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{zj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {x + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} -} \right.}} \\{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times} \\{{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {d_{j} - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {d_{j} - z_{02{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} d\; \tau} -}}\end{matrix}\quad$ $\begin{matrix}{\mspace{115mu} {{- \frac{1}{\left( {\varphi c}_{t} \right)_{j}b}}{\sum\limits_{\iota = {M_{r} + 1}}^{N_{r}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} -} \right.}} \\{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {y - y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {y + y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times}} \\{{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {d_{j} - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{w}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {d_{j} - z_{02{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{xj}\tau}} \right)}} \right\} d\; \tau} -}}\end{matrix}$ $\begin{matrix}{\mspace{115mu} {{- \frac{1}{2\left( {\varphi c}_{t} \right)_{j}{{ab}\left( {d_{j + 1} - d_{j}} \right)}}}{\int_{0}^{t}{\int_{0}^{b}{\int_{0}^{d_{j + 1} - d_{j}}\left\lbrack {{{\psi_{0{yzj}}\left( {v,w,\tau} \right)}\Theta_{3}\left\{ {\frac{\pi x}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} -} \right.}}}}} \\{\left. {{- {\psi_{ayzj}\left( {v,w,\tau} \right)}}\Theta_{4}\left\{ {\frac{\pi x}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right)} \right\rbrack {\Theta_{3}\left( {\frac{\pi w}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}({t - \tau})}}} \right)} \times} \\{{{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {y - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {y + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}}} \right\rbrack {dvdwd}\; \tau} -}}\end{matrix}$ $\begin{matrix}{\mspace{115mu} {{- \frac{1}{2\left( {\varphi c}_{t} \right)_{j}{{ab}\left( {d_{j + 1} - d_{j}} \right)}}}{\int_{0}^{t}{\int_{0}^{a}{\int_{0}^{d_{j + 1} - d_{j}}\left\lbrack {{{\psi_{x0{uj}}\left( {u,w,\tau} \right)}\Theta_{3}\left\{ {\frac{\pi y}{2b},e^{{- {(\frac{\pi y}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} -} \right.}}}}} \\{\left. {{- {\psi_{xbzj}\left( {u,w,\tau} \right)}}{\Theta_{4}\left( {\frac{\pi y}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right)}} \right\rbrack {\Theta_{3}\left( {\frac{\pi w}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}({t - \tau})}}} \right)} \times} \\{{{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {x - u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {x + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}}} \right\rbrack {dudwd}\; \tau} -}}\end{matrix}$ $\begin{matrix}{\mspace{115mu} {{- \frac{1}{4{{ab}\left( {d_{j + 1} - d_{j}} \right)}}}{\int_{0}^{d_{j + 1} - d_{j}}{\int_{0}^{b}{\int_{0}^{a}{{\phi_{j}\left( {u,v,{w + d_{j}}} \right)}\Theta_{3}\left\{ {\frac{\pi w}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}t}} \right) \times}}}}}} \\{{\times \left\lbrack {{\Theta_{3}\left( {\frac{\pi \left( {x - u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}t}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {x + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{zj}t}} \right)}} \right\rbrack \times}} \\{{\times \left\lbrack {{\Theta_{3}\left( {\frac{\pi \left( {y - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}t}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {y + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}t}} \right)}} \right\rbrack {dudvd}w}}\end{matrix}$ *For ι = 1, 2, . . . N_(l) q_(ιj) is flux per unit lengthin layer j and for ι = N_(l) + 1, . . . N_(r) q_(ιj) is flux per unitarea in layer j

Based on the derivation shown in TABLE 2, the general expressions forthe hydrocarbon production occurring through multiple vertical wells,horizontal wells, multiple deviated wells, and multiple fractures in areservoir (e.g., the reservoir (800)) are shown in TABLEs 3 through 7.

TABLE 3The  spatial  average  pressure  response  of  the  line  [z_(02◇j) − z_(01◇j)], ι = ◇, 1 ≤ ◇ ≤ L_(l)  is  given  by.$\quad{\begin{matrix}{p_{j} = {\frac{\left( {d_{j + 1} - d_{j}} \right)}{2\left( {\varphi c}_{t} \right)_{j}{{ab}\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}}{\sum\limits_{\iota = 1}^{L_{t}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {x - x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {x + x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {y - y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {y + y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times}}\end{matrix}{\quad \begin{matrix}{{~~~~~~~~~~~~}{\times \left\{ {{\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{01{\Diamond j}} - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} -} \right.}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} + z_{01{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{01{\Diamond j}} + z_{01{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} -}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} - 2_{02{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} - 2_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} +}} \\{{\left. {{+ {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} + z_{02{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{01{\Diamond j}} + z_{02{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} d\; \tau} +}\end{matrix}}}$ $\begin{matrix}{\mspace{20mu} {{+ \frac{1}{2\left( {\varphi c}_{t} \right)_{j}{b\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}}}{\sum\limits_{\iota = {L_{r} + 1}}^{M_{t}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {y - y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {y + y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{02{\Diamond j}} - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}}\; +} \right.}} \\{\left. {{{+ \Theta_{3}^{\int}}\left\{ {\frac{\pi \left( {z_{02{\Diamond j}} + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{zj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{zj}\tau}} \right)} -} \right.}} \\{{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} d\; \tau} +}\end{matrix}\quad$ $\quad\begin{matrix}{\mspace{25mu} {{+ \frac{1}{2\left( {\varphi c}_{t} \right)_{j}{a\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}}}{\sum\limits_{\iota = {M_{t} + 1}}^{N_{t}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {x - x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {x + x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{02{\Diamond j}} - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}}\; +} \right.}} \\{\left. {{{+ \Theta_{3}^{\int}}\left\{ {\frac{\pi \left( {z_{02{\Diamond j}} + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} -} \right.}} \\{{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} d\; \tau} +}\end{matrix}$ $\begin{matrix}{\mspace{25mu} {{+ \frac{1}{4{\pi \left( {\varphi c}_{t} \right)}_{j}{{ab}\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}}}{\sum\limits_{\iota = {N_{t} + 1}}^{N_{d}}{{U\left( {t - t_{0{\iota j}}} \right)}\mspace{11mu} \sin \mspace{11mu} \vartheta_{0{\iota j}} \times}}}} \\{{\times {\int_{0}^{t - t_{0{\iota j}}}{{q_{j}\left( {t - t_{0{\iota j}} - \tau} \right)}{\int_{z\; 01{\iota j}}^{z_{02{\iota j}}}\left\lbrack \left\{ {{\Theta_{3}\left( {{\frac{\pi}{2a}\left\{ {x - {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{0{\iota j}}\cos \mspace{11mu} \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{a})}^{2}}{\eta {xj}\tau}}} \right)} +} \right. \right.}}}}} \\{\left. {+ {\Theta_{3}\left( {{\frac{\pi}{2a}\left\{ {x + {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{0{\iota j}}\mspace{11mu} \cos \mspace{11mu} \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{a})}^{2}}{\eta {xj}\tau}}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}\left( {{\frac{\pi}{2b}\left\{ {y - {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{0{\iota j}}\sin \mspace{11mu} \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{b})}^{2}}{\eta {yj}\tau}}} \right)} +} \right.}} \\{\left. {+ {\Theta_{3}\left( {{\frac{\pi}{2b}\left\{ {y + {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{0{\iota j}}\mspace{11mu} \sin \mspace{11mu} \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{b})}^{2}}{\eta {yj}\tau}}} \right)}} \right\} \times}\end{matrix}{\quad \begin{matrix}{{~~~~~~~~~~}{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {\eta_{zj}{(\frac{\pi}{d_{j + 1} - d_{j}})}}^{2}}t}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z_{01{\Diamond j}} - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {\eta_{zj}{(\frac{\pi}{d_{j + 1} - d_{j}})}}^{2}}t}} \right)} +} \right.}} \\{{\left. \left. {{+ {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {\eta_{zj}{(\frac{\pi}{d_{j + 1} - d_{j}})}}^{2}}t}} \right)}} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z_{01{\Diamond j}} + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {\eta_{zj}{(\frac{\pi}{d_{j + 1} - d_{j}})}}^{2}}t}} \right)}} \right\} \right\rbrack \mspace{11mu} {dz}_{0{\iota j}}d\; \tau} +}\end{matrix}}$ $\begin{matrix}{\mspace{31mu} {{+ \frac{1}{\left( {\varphi c}_{t} \right)_{j}\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}}{\sum\limits_{\iota = {N_{d} + 1}}^{L_{r}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{zj}\tau}} \right)} -} \right.}} \\{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} -} \right.}} \\{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times}\end{matrix}{\quad \begin{matrix}{{~~~~~~~~~~}{\times \left\{ {{\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{02{\Diamond j}} - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}({t - \tau})}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} +} \right.}} \\{{\left. {{\times \Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{02{\Diamond j}} + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}}} \right\} \mspace{11mu} {d\tau}} +}\end{matrix}}$ $\begin{matrix}{\mspace{25mu} {{+ \frac{1}{\left( {\varphi c}_{t} \right)_{j}{a\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}}}{\sum\limits_{\iota = {L_{r} + 1}}^{M_{r}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {x - x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {x + x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} -} \right.}} \\{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times}\end{matrix}{\quad \begin{matrix}{{~~~~~~~~~}{\times \left\{ {{\Theta_{3}^{\int\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} - {\Theta_{3}^{\int\int}\left( {\frac{\pi \left( {z_{01{\Diamond j}} - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} -} \right.}} \\{{{- {\Theta_{3}^{\int\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} + z_{01{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int\int}\left( {\frac{\pi \left( {z_{01{\Diamond j}} + z_{01{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} -}} \\{{{- {\Theta_{3}^{\int\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} - z_{02{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} +}} \\{{\left. {{+ {\Theta_{3}^{\int\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} + z_{02{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} - {\Theta_{3}^{\int\int}\left( {\frac{\pi \left( {z_{01{\Diamond j}} + z_{02{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \; d\; \tau} +}\end{matrix}}$ $\begin{matrix}{\mspace{34mu} {{+ \frac{1}{\left( {\varphi c}_{t} \right)_{j}{b\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}}}{\sum\limits_{\iota = {M_{r} + 1}}^{N_{r}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} -} \right.}} \\{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {y - y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {y + y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times}}\end{matrix}{\quad \begin{matrix}{{~~~~~~~~~~~}{\times \left\{ {{\Theta_{3}^{\int\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} + {\Theta_{3}^{\int\int}\left( {\frac{\pi \left( {z_{01{\Diamond j}} - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} -} \right.}} \\{{{- {\Theta_{3}^{\int\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} + z_{01{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int\int}\left( {\frac{\pi \left( {z_{01{\Diamond j}} + z_{01{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} -}} \\{{{- {\Theta_{3}^{\int\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} - z_{02{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} +}} \\{{\left. {{+ {\Theta_{3}^{\int\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} + z_{02{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} - {\Theta_{3}^{\int\int}\left( {\frac{\pi \left( {z_{01{\Diamond j}} + z_{02{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \; d\; \tau} +}\end{matrix}}$ $\mspace{25mu} {\begin{matrix}{{+ \frac{1}{2\left( {\varphi c}_{t} \right)_{j}{{ab}\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}}} \times {\int_{0}^{t}{\int_{0}^{a}{\int_{0}^{b}\left\lbrack {{\psi_{j}\left( {u,v,\tau} \right)}\left\{ {{\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{02{\Diamond j}} - d_{j}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}({t - \tau})}}} \right\}} -} \right.} \right.}}}} \\{\left. {{- \Theta_{3}^{\int}}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} - d_{j}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} \right\} -} \\{{{- {\psi_{j + 1}\left( {u,v,\tau} \right)}}\left\{ {{\Theta_{4}^{\int}\left\{ {\frac{\pi \left( {z_{02{\Diamond j}} - d_{j}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}({t - \tau})}}} \right\}} -} \right.}} \\{\left. \left. {{- \Theta_{4}^{\int}}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} - d_{j}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} \right\} \right\rbrack \times}\end{matrix}{\quad \begin{matrix}{{~~~~~~~~~}{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {x - u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {x + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}}} \right\rbrack \times}} \\{{{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {y - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {y + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}}} \right\rbrack {{dudvd}\tau}} +}}\end{matrix}}}$ $\begin{matrix}{\mspace{14mu} {{+ \frac{1}{2\left( {\varphi c}_{t} \right)_{j}{{ab}\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}}} \times}} \\{{\times {\int_{0}^{t}{\int_{0}^{b}{\int_{0}^{d_{j + 1} - d_{j}}{\left\{ {{{\psi_{0{yzj}}\left( {v,w,\tau} \right)}{\Theta_{3}\left( {\frac{\pi x}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}({t - \tau})}}} \right)}} - {{\psi_{ayzj}\left( {v,w,\tau} \right)}{\Theta_{4}\left( {\frac{\pi x}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}({t - \tau})}}} \right)}}} \right\} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}\left\{ {\frac{\pi\left( {y - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}({t - \tau})}}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi\left( {y + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{02{\Diamond j}} - d_{j} - w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}({t - \tau})}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} - d_{j} - w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} +} \right.}} \\{{\left. {{\times \Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{02{\Diamond j}} - d_{j} + w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} - d_{j} + w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}}} \right\} {dvdwd\tau}} +}\end{matrix}\quad$ $\begin{matrix}{\mspace{14mu} {{+ \frac{1}{2\left( {\varphi c}_{t} \right)_{j}{{ab}\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}}} \times}} \\{{\times {\int_{0}^{t}{\int_{0}^{a}{\int_{0}^{d_{j + 1} - d_{j}}{\left\{ {{{\psi_{x\; 0{zj}}\left( {v,w,\tau} \right)}{\Theta_{3}\left( {\frac{\pi y}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}({t - \tau})}}} \right)}} - {{\psi_{xbzj}\left( {u,w,\tau} \right)}{\Theta_{4}\left( {\frac{\pi y}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}({t - \tau})}}} \right)}}} \right\} \times}}}}}} \\{{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi\left( {x - u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}({t - \tau})}}} \right\}} + {\Theta_{3}\left( {\frac{\pi\left( {x + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}}} \right\rbrack \times}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{02{\Diamond j}} - d_{j} - w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}({t - \tau})}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} - d_{j} - w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} +} \right.}} \\{{\left. {{\times \Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{02{\Diamond j}} - d_{j} + w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} - d_{j} + w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}}} \right\} {dvdwd\tau}} +}\end{matrix}\quad$ $\begin{matrix}{\mspace{14mu} {{+ \frac{1}{4{{ab}\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}_{j}}} \times}} \\{{\times {\int_{0}^{d_{j + 1} - d_{j}}{\int_{0}^{b}{\int_{0}^{a}{\phi_{j}\left\{ {\left( {u,v,{w + d_{j}}} \right)\left\lbrack {\left\{ {{\Theta_{3}\left( {\frac{\pi\left( {x - u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}t}} \right)} + {\Theta_{3}\left( {\frac{\pi\left( {x + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}t}} \right)}} \right\} \times} \right.} \right.}}}}}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi\left( {y - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}t}} \right)} + {\Theta_{3}\left( {\frac{\pi\left( {y + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}t}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} - d_{j} - w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}t}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z_{01{\Diamond j}} - d_{j} - w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}t}} \right\}} +} \right.}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} - d_{j} + w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}t}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z_{01{\Diamond j}} - d_{j} + w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}}} \right\}}} \right\} {dudvdw}}}\end{matrix}\quad$

TABLE 4The  spatial  average  pressure  response  of  the  line  [x_(02◇j) − x_(01◇j)], ι = ◇, L_(l + 1) ≤ ◇ ≤ M_(l)  is  given  by.$\begin{matrix}{\quad\begin{matrix}{p_{j} = {\frac{1}{2\left( {\varphi c}_{t} \right)_{j}\mspace{11mu} {b\left( {x_{02{\Diamond j}} - x_{01{\Diamond j}}} \right)}}\mspace{11mu} {\sum\limits_{\iota = 1}^{L_{t}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} - x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} +} \right.}} \\{\left. {{+ {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} + x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} + x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {y - y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {y + y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {z - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z + z_{01{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} -} \right.}} \\{{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z - z_{02{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z + z_{02{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \; d\; \tau} +}\end{matrix}} & (0.10) \\{\quad\begin{matrix}{\mspace{34mu} {{+ \frac{a}{2\left( {\varphi c}_{t} \right)_{j}{b\left( {d_{j + 1} - d_{j}} \right)}\left( {x_{02{\Diamond j}} - x_{01{\Diamond j}}} \right)}}{\sum\limits_{\iota = {L_{\iota} + 1}}^{M_{t}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {y - y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {y + y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {z - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {z + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{wj}\tau}} \right)} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} -} \right.}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} -}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} - x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} - x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} +}} \\{{\left. {{+ {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} + x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} + x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} d\; \tau} +}\end{matrix}} & \;\end{matrix}$ $\; {\quad{\begin{matrix}{\mspace{20mu} {{+ \frac{1}{2\left( {\varphi c}_{t} \right)_{j}\left( {d_{j + 1} - d_{j}} \right)\left( {x_{02{\Diamond j}} - x_{01{\Diamond j}}} \right)}}{\sum\limits_{\iota = {M_{\iota} + 1}}^{N_{t}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} - x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} - x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} +} \right.}} \\{\left. {{+ {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} + x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} + x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {z - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {z + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} -} \right.}} \\{{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} d\; \tau} +}\end{matrix}\quad}}$ $\begin{matrix}{\mspace{25mu} {{+ \frac{1}{4{\pi \left( {\varphi c}_{t} \right)}_{j}{b\left( {d_{j + 1} - d_{j}} \right)}\left( {x_{02{\Diamond j}} - x_{01{\Diamond j}}} \right)}}{\sum\limits_{\iota = {N_{t} + 1}}^{N_{d}}{{U\left( {t - t_{0{\iota j}}} \right)}\mspace{11mu} \sin \mspace{11mu} \vartheta_{0{\iota j}} \times}}}} \\{{\times {\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)}{\int_{z\; 01{\iota j}}^{z_{02{\iota j}}}\left\lbrack \left\{ {{\Theta_{3}\left( {{\frac{\pi}{2b}\left\{ {y - {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{0{\iota j}}\cos \mspace{11mu} \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{b})}^{2}}{\eta {yj}\tau}}} \right)} +} \right. \right.}}}}} \\{\left. {+ {\Theta_{3}\left( {{\frac{\pi}{2b}\left\{ {y - {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\iota j}}\;}\cos \mspace{11mu} \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{b})}^{2}}{\eta {yj}\tau}}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {{\frac{\pi}{2a}\left\{ {x_{02{\Diamond j}} - {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{0{\iota j}}\mspace{11mu} \sin \mspace{11mu} \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{a})}^{2}}{\eta {xj}\tau}}} \right)} -} \right.}} \\{{{- {\Theta_{3}^{\int}\left( {{\frac{\pi}{2a}\left\{ {x_{01{\Diamond j}} - {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{0{\iota j}}\mspace{11mu} \sin \mspace{11mu} \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{a})}^{2}}{\eta {xj}\tau}}} \right)}} +}} \\{{{+ {\Theta_{3}^{\int}\left( {{\frac{\pi}{2a}\left\{ {x_{02{\Diamond j}} + {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{0{\iota j}}\mspace{11mu} \sin \mspace{11mu} \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{a})}^{2}}{\eta {xj}\tau}}} \right)}} -}} \\{\left. {- {\Theta_{3}^{\int}\left( {{\frac{\pi}{2a}\left\{ {x_{01{\Diamond j}} + {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{0{\iota j}}\mspace{11mu} \sin \mspace{11mu} \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{a})}^{2}}{\eta {xj}\tau}}} \right)}} \right\} \times} \\{{\left. {\times \left\{ {{\Theta_{3}\left( {{\frac{\pi}{2d}\left( {z - z_{0{\iota j}}} \right)},e^{{- {(\frac{\pi}{d})}^{2}}{\eta {yj}\tau}}} \right)} + {\Theta_{3}\left( {{\frac{\pi}{2d}\left( {z + z_{0{\iota j}}} \right)},e^{{- {(\frac{\pi}{d})}^{2}}{\eta {yj}\tau}}} \right)}} \right\}} \right\rbrack \; {dz}_{0{\iota j}}d\; \tau} +}\end{matrix}\quad$ $\; {\begin{matrix}{\mspace{20mu} {{+ \frac{a}{\left( {\varphi c}_{t} \right)_{j}\left( {d_{j + 1} - d_{j}} \right)\left( {x_{02{\Diamond j}} - x_{01{\Diamond j}}} \right)}}{\sum\limits_{\iota = {N_{d} + 1}}^{L_{r}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} -} \right.}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} -}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} - x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} - x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} +}} \\{\left. {{+ {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} + x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} + x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} -} \right.}} \\{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times} \\{{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {z - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {z + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \; d\; \tau} +}}\end{matrix}\quad}$ $\begin{matrix}{\mspace{25mu} {{+ \frac{1}{\left( {\varphi c}_{t} \right)_{j}\left( {x_{02{\Diamond j}} - x_{01{\Diamond j}}} \right)}}{\sum\limits_{\iota = {L_{r} + 1}}^{M_{r}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} - x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} - x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} +} \right.}} \\{\left. {{+ {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} + x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} + x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{yj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} -} \right.}} \\{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y - y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y + y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {z - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z + z_{01{\Diamond j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} -} \right.}} \\{{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z - z_{02{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z + z_{02{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \; d\; \tau} +}\end{matrix}\quad$ $\begin{matrix}{\mspace{25mu} {{+ \frac{a}{\left( {\varphi c}_{t} \right)_{j}{b\left( {x_{02{\Diamond j}} - x_{01{\Diamond j}}} \right)}}}{\sum\limits_{\iota = {M_{r} + 1}}^{N_{r}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} -} \right.}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} -}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} - x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} + x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} +}} \\{\left. {{+ {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} + x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} + x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {y - y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {y + y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times}}\end{matrix}\begin{matrix}{{~~~~~~}{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {z - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z + z_{01{\Diamond j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} -} \right.}} \\{{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z - z_{02{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z + z_{02{\Diamond j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \; d\; \tau} +}\end{matrix}$ $\mspace{25mu} \begin{matrix}{{+ \frac{1}{2\left( {\varphi c}_{t} \right)_{j}{b\left( {d_{j + 1} - d_{j}} \right)}\left( {x_{02{\Diamond j}} - x_{01{\Diamond j}}} \right)}} \times {\int_{0}^{t}{\int_{0}^{a}{\int_{0}^{b}\left\{ {{{\psi_{j}\left( {u,v,\tau} \right)}\Theta_{3}\left\{ {\frac{\pi \left( {z - d_{j}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}({t - \tau})}}} \right\}} -} \right.}}}} \\{\left. {{- {\psi_{j + 1}\left( {u,v,\tau} \right)}}\Theta_{4}\left\{ {\frac{\pi \left( {z - d_{j}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} \right\rbrack \times} \\{{\times \left\lbrack {{\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {x_{02{\Diamond j}} - u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {x_{01{\Diamond j}} - u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}} +} \right.}} \\{\left. {{{+ \Theta_{3}^{\int}}\left\{ {\frac{\pi \left( {x_{02{\Diamond j}} + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {x_{01{\Diamond j}} + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}}} \right\rbrack \times} \\{{{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {y - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {y + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}}} \right\rbrack \mspace{11mu} {dudvd\tau}} +}}\end{matrix}$ $\mspace{25mu} \begin{matrix}{{+ \frac{1}{2\left( {\varphi c}_{t} \right)_{j}{b\left( {d_{j + 1} - d_{j}} \right)}\left( {x_{02{\Diamond j}} - x_{01{\Diamond j}}} \right)}} \times} \\{{\times {\int_{0}^{t}{\int_{0}^{b}{\int_{0}^{d_{j + 1} - d_{j}}\left\{ {{{\psi_{0{{yz}j}}\left( {v,w,\tau} \right)}\left\{ {{\Theta_{3}^{\int}\left( {\frac{{\pi x}_{02{\Diamond j}}}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}({t - \tau})}}} \right)} - {\Theta_{3}^{\int}\left( {\frac{{\pi x}_{01{\Diamond j}}}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}({t - \tau})}}} \right)}} \right\}} -} \right.}}}}} \\{\left. {{- \psi_{ayzj}}\mspace{11mu} \left( {v,w,\tau} \right)\left\{ {{\Theta_{4}^{\int}\left( {\frac{{\pi x}_{02{\Diamond j}}}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}({t - \tau})}}} \right)} - {\Theta_{4}^{\int}\left( {\frac{{\pi x}_{01{\Diamond j}}}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}({t - \tau})}}} \right)}} \right\}} \right\} \times} \\{{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {y - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {y + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}}} \right\rbrack \; \times}} \\{{{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {z - d_{j\;} - w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {z - d_{j\;} + w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}}} \right\rbrack \; {dudwd}\; \tau} +}}\end{matrix}$ $\; {\begin{matrix}{\mspace{14mu} {{+ \frac{1}{2\left( {\varphi c}_{t} \right)_{j}{b\left( {d_{j + 1} - d_{j}} \right)}\left( {x_{02{\Diamond j}} - x_{01{\Diamond j}}} \right)}} \times}} \\{{\times {\int_{0}^{t}{\int_{0}^{a}{\int_{0}^{d_{j + 1} - d_{j}}\left\{ {{{\psi_{x\; 0{zj}}\left( {u,w,\tau} \right)}{\Theta_{3}\left( {\frac{\pi y}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}({t - \tau})}}} \right)}} - {{\psi_{xbzj}\left( {u,w,\tau} \right)}{\Theta_{4}\left( {\frac{\pi y}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}({t - \tau})}}} \right)}}} \right\}}}}}} \\{{\times \left\lbrack {{\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {x_{02{\Diamond j}} - u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {x_{01{\Diamond j}} - u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}} +} \right.}} \\{\left. {{{+ \Theta_{3}^{\int}}\left\{ {\frac{\pi \left( {x_{02{\Diamond j}} + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {x_{01{\Diamond j}} + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}}} \right\rbrack \times} \\{{{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {z - d_{j\;} - w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {z - d_{j\;} + w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}}} \right\rbrack \; {dudwd}\; \tau} +}}\end{matrix}\quad}$ $\mspace{25mu} \begin{matrix}{{+ \frac{1}{4\; \left( {d_{j + 1} - d_{j}} \right)\left( {x_{02{\Diamond j}} - x_{01{\Diamond j}}} \right)}} \times {\int_{0}^{d_{j + 1} - d_{j}}{\int_{0}^{b}{\int_{0}^{a}{{\phi_{j}\left( {u,v,{w + d_{j}}} \right)} \times}}}}} \\{{\times \left\lbrack {{\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {x_{02{\Diamond j}} - u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {x_{01{\Diamond j}} - u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}} +} \right.}} \\{\left. {{{+ \Theta_{3}^{\int}}\left\{ {\frac{\pi \left( {x_{02{\Diamond j}} + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {x_{01{\Diamond j}} + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}}} \right\rbrack \times} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {y - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}t}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {y + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}t}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}\left\{ {\frac{\pi \left( {z - d_{j\;} - w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}t}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {z - d_{j\;} + w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}t}} \right\}}} \right\} {dudvdw}}}\end{matrix}$

TABLE 5The  spatial  average  pressure  response  of  the  inclined  line [(z_(02◇j) − z_(01◇j)) sin   ϑ_(0◇j)], ι = ◇j, N_(l) + 1 ≤ ◇ ≤ N_(d)  is  given  by$\begin{matrix}\begin{matrix}{p = {\frac{1}{4{{\pi a}b}\mspace{11mu} \left( {\varphi c}_{t} \right)_{j}\; \left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)} \times}} \\{{\times {\sum\limits_{\iota = 1}^{L_{1}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}\left\lbrack {{q_{\iota \; j}\left( {t - t_{0{\iota j}} - \tau} \right)}{\int_{z_{01{\Diamond j}}}^{z_{02{\Diamond j}}}\left\{ {{\Theta_{3}\left( {{\frac{\pi}{2a}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}} - x_{0{\iota j}}} \right)},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} +} \right.}} \right.}}}}} \\{\left. {{+ \Theta_{3}}\; \left( {{\frac{\pi}{2a}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}} + x_{0{\iota j}}} \right)},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} \right\} \times} \\{{\times \left\{ {{\Theta_{3}\; \left( {{\frac{\pi}{2b}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\sin \mspace{11mu} \theta_{0{\Diamond j}}} - y_{0\iota \mspace{11mu} j}} \right)},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +} \right.}} \\{\left. {{+ \Theta_{3}}\; \left( {{\frac{\pi}{2b}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\sin \mspace{11mu} \theta_{0{\Diamond j}}} + y_{0\iota \mspace{11mu} j}} \right)},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} \right\} \times}\end{matrix} & (0.11) \\\begin{matrix}{{~~~~~~~~~}{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {z - z_{01\iota \mspace{11mu} j}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z + z_{01{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} +} \right.}} \\{{\left. {\left. {{+ {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z - z_{02\iota \mspace{11mu} j}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z + z_{02{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \mspace{11mu} {dz}} \right\rbrack d\; \tau} +}\end{matrix} & \;\end{matrix}$ $\begin{matrix}{\mspace{40mu} {{+ \frac{1}{4{\pi b}\mspace{11mu} \left( {d_{j + 1} - d_{j}} \right)\left( {\varphi c}_{t} \right)_{j}\; \left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}} \times}} \\{{\times {\sum\limits_{\iota = {L_{l} + 1}}^{M_{l}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}\left\lbrack {{q_{\iota \; j}\left( {t - t_{0{\iota j}} - \tau} \right)}{\int_{z_{01{\Diamond j}}}^{z_{02{\Diamond j}}}\left\{ {{\Theta_{3}\left( {\frac{\pi \left( {z - z_{0\iota \mspace{11mu} j}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} +} \right.}} \right.}}}}} \\{\left. {{+ \Theta_{3}}\; \left( {\frac{\pi \left( {z + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} \right\} \times}\end{matrix}$ $\begin{matrix}{{~~~~~~~~~~~~}{\times \left\{ {{\Theta_{3}\; \left( {{\frac{\pi}{2b}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\sin \mspace{11mu} \theta_{0{\Diamond j}}} - y_{0\iota \mspace{11mu} j}} \right)},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +} \right.}} \\{\left. {{+ \Theta_{3}}\; \left( {{\frac{\pi}{2b}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\sin \mspace{11mu} \theta_{0{\Diamond j}}} + y_{0\iota \mspace{11mu} j}} \right)},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int}\; \left( {{\frac{\pi}{2a}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}} - x_{01{\iota j}}} \right)},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} +} \right.}} \\{{{{+ \Theta_{3}^{\int}}\; \left( {{\frac{\pi}{2a}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}} + x_{01\iota \mspace{11mu} j}} \right)},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} +}} \\{{{+ \Theta_{3}^{\int}}\; \left( {{\frac{\pi}{2a}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}} - x_{02\iota \mspace{11mu} j}} \right)},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \\{{\left. {\left. {{+ \Theta_{3}^{\int}}\; \left( {{\frac{\pi}{2a}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}} + x_{02\iota \mspace{11mu} j}} \right)},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} \right\} \mspace{11mu} d\; z} \right\rbrack d\; \tau} +}\end{matrix}\quad$ $\begin{matrix}{\mspace{11mu} {{+ \frac{1}{4{\pi a}\mspace{11mu} \left( {d_{j + 1} - d_{j}} \right)\left( {\varphi c}_{t} \right)_{j}\; \left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}}\mspace{11mu} {\sum\limits_{\iota = {M_{l} + 1}}^{N_{l}}{{U\left( {t - t_{0{\iota j}}} \right)} \times}}}} \\{{\times {\int_{0}^{t - t_{0{\iota j}}}\left\lbrack {{q_{\iota}\left( {t - t_{0{\iota j}} - \tau} \right)}{\int_{z_{01{\Diamond j}}}^{z_{02{\Diamond j}}}\left\{ {{\Theta_{3}\left( {{\frac{\pi}{2a}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{14mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}} - x_{0{\iota j}}} \right)},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{zj}\tau}} \right)} +} \right.}} \right.}}} \\{\left. {{+ \Theta_{3}}\; \left( {{\frac{\pi}{2a}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}} + x_{0{\iota j}}} \right)},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} \right\} \times} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {z - z_{0\iota \; j}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {z + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \times}}\end{matrix}{\quad \begin{matrix}{{~~~~~~~}{\times \left\{ {{\Theta_{3}^{\int}\; \left( {{\frac{\pi}{2b}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\sin \mspace{11mu} \theta_{0{\Diamond j}}} - y_{01\iota \; j}} \right)},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +} \right.}} \\{{{{+ \Theta_{3}^{\int}}\; \left( {{\frac{\pi}{2b}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\sin \mspace{11mu} \theta_{0{\Diamond j}}} + y_{01\iota \; j}} \right)},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +}} \\{{{{+ \Theta_{3}^{\int}}\; \left( {{\frac{\pi}{2b}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\sin \mspace{11mu} \theta_{0{\Diamond j}}} - y_{02\iota \; j}} \right)},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +}} \\{{\left. {\left. {{+ \Theta_{3}^{\int}}\; \left( {{\frac{\pi}{2b}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\sin \mspace{11mu} \theta_{0{\Diamond j}}} + y_{02\iota \; j}} \right)},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} \right\} \mspace{11mu} {dz}} \right\rbrack \; d\; \tau} +}\end{matrix}}$ $\begin{matrix}{\mspace{11mu} {{+ \frac{1}{8\left( {\varphi c}_{t} \right)_{j}{{ab}\left( {d_{j + 1} - d_{j}} \right)}\; \left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}} \times}} \\{{\times {\sum\limits_{\iota = {N_{l} + 1}}^{N_{d}}{{U\left( {t - t_{0{\iota j}}} \right)}\mspace{11mu} \sin \mspace{11mu} \vartheta_{0\iota \; j}{\int_{0}^{t - t_{0{\iota j}}}{q_{\iota}\; \left( {t - t_{0{\iota j}} - \tau} \right) \times}}}}}} \\{{\times {\int_{z_{01{\iota j}}}^{z_{02{\iota j}}}{\int_{z_{01{\Diamond j}}}^{z_{02{\Diamond j}}}\left\lbrack \left\{ {{\Theta_{3}\left( {{\frac{\pi}{2a}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}} - {\left( {z_{0{\iota j}} - \gamma_{0\iota \mspace{11mu} j}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\iota j}}\;}\cos \mspace{11mu} \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} +} \right. \right.}}}} \\{\left. {{+ \Theta_{3}}\; \left( {{\frac{\pi}{2a}\left\{ {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}} + {\left( {z_{0{\iota j}} - \gamma_{0\iota \; j}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\iota j}}\;}\cos \mspace{11mu} \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} \right\} \times} \\{{\times \left\{ {{\Theta_{3}\; \left( {{\frac{\pi}{2b}\left\{ {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\sin \mspace{11mu} \theta_{0{\Diamond j}}} - {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\iota j}}\;}\sin \mspace{11mu} \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +} \right.}} \\{\left. {{+ \Theta_{3}}\; \left( {{\frac{\pi}{2b}\left\{ {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\sin \mspace{11mu} \theta_{0{\Diamond j}}} + {\left( {z_{0{\iota j}} - \gamma_{0\iota \; j}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\iota j}}\;}\sin \mspace{11mu} \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} \right\} \times} \\{{\left. {\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {z - z_{0\iota \; j}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{yj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {z + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{yj}\tau}} \right)}} \right\}} \right\rbrack \mspace{11mu} {dzdz}_{0\iota \; j}d\; \tau} +}\end{matrix}\quad$ $\begin{matrix}{\mspace{20mu} {{+ \frac{1}{2{\pi^{2}\left( {d_{j + 1} - d_{j}} \right)}\left( {\varphi c}_{t} \right)_{j}\; \left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}}\mspace{11mu} {\sum\limits_{\iota = {N_{d} + 1}}^{L_{r}}{{U\left( {t - t_{0{\iota j}}} \right)} \times}}}} \\{{\times {\int_{0}^{t - t_{0{\iota j}}}\left\lbrack {{q_{\iota \; j}\left( {t - t_{0{\iota j}} - \tau} \right)}{\int_{z_{01{\Diamond j}}}^{z_{02{\Diamond j}}}\left\{ {{\Theta_{3}\left( {\frac{\pi \left( {z - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} +} \right.}} \right.}}} \\{\left. {{+ \Theta_{3}}\; \left( {\frac{\pi \left( {z + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} \right\} \times}\end{matrix}$ $\begin{matrix}{{~~~~~~~~~}{\times \left\{ {{\Theta_{3}^{\int}\left( {{\frac{\pi}{2b}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\sin \mspace{11mu} \theta_{0{\Diamond j}}} - y_{01\iota}} \right)},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +} \right.}} \\{{{{+ \Theta_{3}^{\int}}\; \left( {{\frac{\pi}{2b}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\sin \mspace{11mu} \theta_{0{\Diamond j}}} + y_{01\iota}} \right)},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +}} \\{{{{+ \Theta_{3}^{\int}}\; \left( {{\frac{\pi}{2b}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\sin \mspace{11mu} \theta_{0{\Diamond j}}} - y_{02\iota}} \right)},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +}} \\{\left. {{+ \Theta_{3}^{\int}}\; \left( {{\frac{\pi}{2b}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\sin \mspace{11mu} \theta_{0{\Diamond j}}} + y_{02\iota}} \right)},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int}\; \left( {{\frac{\pi}{2a}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}} - x_{01\iota}} \right)},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{zj}\tau}} \right)} +} \right.}} \\{{{{+ \Theta_{3}^{\int}}\; \left( {{\frac{\pi}{2a}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}} + x_{01\iota}} \right)},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} +}} \\{{{{+ \Theta_{3}^{\int}}\; \left( {{\frac{\pi}{2a}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}} - x_{02\iota}} \right)},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{zj}\tau}} \right)} +}} \\{{\left. {\left. {{+ \Theta_{3}^{\int}}\; \left( {{\frac{\pi}{2a}\left( {{\left( {z - \gamma_{{0{\Diamond j}}\mspace{14mu}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}} + x_{02\iota}} \right)},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{zj}\tau}} \right)} \right\} \mspace{11mu} {dz}} \right\rbrack \; d\; \tau} +}\end{matrix}$ $\begin{matrix}{\mspace{11mu} {{+ \frac{1}{2\pi^{2}{a\left( {\varphi c}_{t} \right)}_{j}\; \left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}}\mspace{11mu} {\sum\limits_{\iota = {L_{r} + 1}}^{M_{r}}{{U\left( {t - t_{0{\iota j}}} \right)} \times}}}} \\{{\times {\int_{0}^{t - t_{0{\iota j}}}\left\lbrack {{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)}{\int_{z_{01{\Diamond j}}}^{z_{02{\Diamond j}}}\left\{ {{\Theta_{3}\left( {{\frac{\pi}{2a}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{14mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}} - x_{0{\iota j}}} \right)},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} +} \right.}} \right.}}} \\{\left. {{+ \Theta_{3}}\; \left( {{\frac{\pi}{2a}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}} + x_{0{\iota j}}} \right)},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int}\; \left( {{\frac{\pi}{2b}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\sin \mspace{11mu} \theta_{0{\Diamond j}}} - y_{01\iota}} \right)},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +} \right.}} \\{{{{+ \Theta_{3}^{\int}}\; \left( {{\frac{\pi}{2b}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\sin \mspace{11mu} \theta_{0{\Diamond j}}} + y_{01\iota}} \right)},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +}} \\{{{{+ \Theta_{3}^{\int}}\; \left( {{\frac{\pi}{2b}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\sin \mspace{11mu} \theta_{0{\Diamond j}}} - y_{02\iota}} \right)},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +}} \\{\left. {{+ \Theta_{3}^{\int}}\; \left( {{\frac{\pi}{2b}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\sin \mspace{11mu} \theta_{0{\Diamond j}}} + y_{02\iota}} \right)},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} \right\} \times}\end{matrix}{\quad \begin{matrix}{{~~~~~~~}{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {z - z_{01\iota}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z + z_{01{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} +} \right.}} \\{{\left. {\left. {{+ {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z - z_{02\iota}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z + z_{02{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \mspace{11mu} {dz}} \right\rbrack d\; \tau} +}\end{matrix}}$ $\begin{matrix}{\mspace{14mu} {{+ \frac{1}{2\pi^{2}{b\left( {\varphi c}_{t} \right)}_{j}\; \left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}} \times}} \\{{\times {\sum\limits_{\iota = {M_{r} + 1}}^{N_{r}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}\left\lbrack {{q_{\iota}\left( {t - t_{0{\iota j}} - \tau} \right)}{\int_{z_{01{\Diamond j}}}^{z_{02{\Diamond j}}}\left\{ {{\Theta_{3}\left( {{\frac{\pi}{2b}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\sin \mspace{11mu} \theta_{0{\Diamond j}}} - y_{0{\iota j}}} \right)},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +} \right.}} \right.}}}}} \\{\left. {{+ \Theta_{3}}\; \left( {{\frac{\pi}{2b}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\sin \mspace{11mu} \theta_{0{\Diamond j}}} + y_{0{\iota j}}} \right)},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int}\; \left( {{\frac{\pi}{2a}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\sin \mspace{11mu} \theta_{0{\Diamond j}}} - x_{01\iota}} \right)},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{zj}\tau}} \right)} +} \right.}} \\{{{{+ \Theta_{3}^{\int}}\; \left( {{\frac{\pi}{2a}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}} + x_{01\iota}} \right)},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{zj}\tau}} \right)} +}} \\{{{{+ \Theta_{3}^{\int}}\; \left( {{\frac{\pi}{2a}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}} - x_{02\iota}} \right)},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} +}} \\{\left. {{+ \Theta_{3}^{\int}}\; \left( {{\frac{\pi}{2a}\left( {{\left( {z - \gamma_{0{\Diamond j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}} + x_{02\iota}} \right)},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} \right\} \times}\end{matrix}{\quad \begin{matrix}{{~~~~~~~~}{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {z - z_{01\iota}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z + z_{01{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} +} \right.}} \\{{\left. {\left. {{+ {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z - z_{02\iota}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z + z_{02{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \mspace{11mu} {dz}} \right\rbrack d\; \tau} +}\end{matrix}}$ $\begin{matrix}{\mspace{14mu} {{{+ \frac{4}{\left( {\varphi c}_{t} \right)_{j}{{ab}\left( {d_{j + 1} - d_{j}} \right)}\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}} \times {\sum\limits_{m = 0}^{\infty}\sum\limits_{l = 0}^{\infty}}} \ni_{m} \ni_{l}{\int_{z_{01{\Diamond j}}}^{z_{02{\Diamond j}}}\; {{\cos \left( \frac{m\; \pi \mspace{11mu} \left( {z - \gamma_{0\Diamond \; j}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\sin \mspace{11mu} \theta_{0{\Diamond j}}}{b} \right)}\mspace{11mu} \cos \mspace{11mu} \left( \frac{l\; {\pi \left( {z - d_{j}} \right)}}{\left( {d_{j + 1} - d_{j}} \right)} \right) \times}}}} \\{{{\times {\int_{0}^{t}\; \left\{ {{\overset{=}{\psi}}_{0{yz}}\mspace{11mu} \left( {m,l,\tau} \right)\mspace{11mu} \Theta_{3}\mspace{11mu} \left( \frac{\pi \mspace{11mu} \left( {z - \gamma_{0\Diamond \; j}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}}{2a} \right)\mspace{11mu} e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right)}} -}} \\{{\left. {{- {\overset{=}{\psi}}_{ayz}}\mspace{11mu} \left( {m,l,\tau} \right)\mspace{11mu} \Theta_{4}\; \left( {\frac{\pi \mspace{11mu} \left( {z - \gamma_{0\Diamond \; j}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right)} \right\} \mspace{11mu} e^{{- {\{{{{(\frac{m\; \pi}{b})}^{2}\eta_{yj}} + {{(\frac{l\; \pi}{d_{j + 1} - d_{j}})}^{2}\eta_{zj}}}\}}}{({t - \tau})}}d\; {\tau dz}} +}\end{matrix}$ $\begin{matrix}{\mspace{14mu} {{{+ \frac{4}{\left( {\varphi c}_{t} \right)_{j}{{ab}\left( {d_{j + 1} - d_{j}} \right)}\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}} \times {\sum\limits_{n = 0}^{\infty}\sum\limits_{l = 0}^{\infty}}} \ni_{n} \ni_{l}{\int_{z_{01{\Diamond j}}}^{z_{02{\Diamond j}}}\; {{\cos \left( \frac{n\; \pi \mspace{11mu} \left( {z - \gamma_{0\Diamond \; j}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}}{a} \right)}\mspace{11mu} \cos \mspace{11mu} \left( \frac{l\; {\pi \left( {z - d_{j}} \right)}}{d_{j + 1} - d_{j}} \right) \times}}}} \\{{{\times {\int_{0}^{t}\; \left\{ {{\overset{=}{\psi}}_{x\; 0z}\mspace{11mu} \left( {n,l,\tau} \right)\mspace{11mu} \Theta_{3}\mspace{11mu} \left( \frac{\pi \mspace{11mu} \left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)\mspace{11mu} \sin \mspace{11mu} \vartheta_{0{\Diamond j}}}{2b} \right)\mspace{11mu} e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right)}} -}} \\{{\left. {{- {\overset{=}{\psi}}_{xbz}}\mspace{11mu} \left( {n,l,\tau} \right)\mspace{11mu} \Theta_{4}\; \left( {\frac{\pi \mspace{11mu} \left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)\mspace{11mu} \sin \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right)} \right\} \mspace{11mu} e^{{- {\{{{{(\frac{n\; \pi}{a})}^{2}\eta_{xj}} + {{(\frac{l\; \pi}{d_{j + 1} - d_{j}})}^{2}\eta_{zj}}}\}}}{({t - \tau})}}d\; {\tau dz}} +}\end{matrix}$ $\begin{matrix}{\mspace{14mu} {{+ \frac{4}{\left( {\varphi c}_{t} \right)_{j}{{ab}\left( {d_{j + 1} - d_{j}} \right)}\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}} \times}} \\{\; {{\times \; {\sum\limits_{n = 0}^{\infty}\sum\limits_{m = 0}^{\infty}}} \ni_{n} \ni_{m}{\int_{z_{01{\Diamond j}}}^{z_{02{\Diamond j}}}\; {\cos \mspace{11mu} \left( \frac{n\; \pi \mspace{11mu} \left( {z - \gamma_{0\Diamond \; j}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}}{a} \right)\mspace{11mu} \cos \mspace{11mu} \left( \frac{m\; \pi \mspace{11mu} \left( {z - \gamma_{0\Diamond \; j}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}}{b} \right) \times}}}} \\{{\times {\int_{0}^{t}\; \left\{ {{{\overset{=}{\psi}}_{{xy}\; 0}\mspace{11mu} \left( {n,m,\tau} \right)\mspace{11mu} \Theta_{3}\; \left( {\frac{\pi \left( {z - d_{j}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right)} -} \right.}}} \\{{\left. {{- {\overset{=}{\psi}}_{xyd}}\mspace{11mu} \left( {n,m,\tau} \right)\mspace{11mu} \Theta_{4}\; \left( {\frac{\pi \left( {z - d_{j}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right)} \right\} \mspace{11mu} e^{{- {\{{{{(\frac{n\; \pi}{a})}^{2}\eta_{xj}} + {{(\frac{m\; \pi}{b})}^{2}\eta_{yj}}}\}}}{({t - \tau})}}d\; {\tau dz}} +}\end{matrix}{\quad \begin{matrix}{\mspace{20mu} {{+ \frac{1}{8{{ab}\left( {d_{j + 1} - d_{j}} \right)}\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}}\; {\int_{0}^{d}{\int_{0}^{b}{\int_{0}^{a}{{\phi_{j}\left( {u,v,w,{+ d_{j}}} \right)} \times}}}}}} \\{{\times {\int_{z_{01{\Diamond j}}}^{z_{02{\Diamond j}}}\left\lbrack \left\{ {{\Theta_{3}\; \left( {{\frac{\pi}{2a}\left( {{\left( {z - \gamma_{{0{\Diamond j}}\;}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}} - u} \right)},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}t}} \right)} +} \right. \right.}}} \\{\left. {{+ \Theta_{3}}\; \left( {{\frac{\pi}{2a}\left( {{\left( {z - \gamma_{{0{\Diamond j}}\;}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\cos \mspace{11mu} \theta_{0{\Diamond j}}} + u} \right)},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}t}} \right)} \right\} \times} \\{{\times \left\{ {{\Theta_{3}\; \left( {{\frac{\pi}{2b}\left( {{\left( {z - \gamma_{{0{\Diamond j}}\;}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\sin \mspace{11mu} \theta_{0{\Diamond j}}} - v} \right)},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}t}} \right)} +} \right.}} \\{\left. {{+ \Theta_{3}}\; \left( {{\frac{\pi}{2b}\left( {{\left( {z - \gamma_{{0{\Diamond j}}\;}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\Diamond j}}\;}\sin \mspace{11mu} \theta_{0{\Diamond j}}} + v} \right)},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}t}} \right)} \right\} \times} \\{\left. {\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {z - d_{j\;} - w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}t}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {z - d_{j\;} + w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}t}} \right)}} \right\}} \right\rbrack \mspace{11mu} {dudvdwdz}}\end{matrix}}$

TABLE 6The  spatial  average  pressure  response  of  the  rectangle  [(x_(02◇j) − x_(01◇j))  (y_(02◇j) − y_(01◇j))], ι = ◇, N_(l) + 1 ≤ ◇ ≤ L_(r)  is  given  by$\begin{matrix}\begin{matrix}{p_{j} = {\frac{1}{\left( {\varphi c}_{t} \right)_{j}\left( {x_{02{\Diamond j}} - x_{01{\Diamond j}}} \right)\left( {y_{02{\Diamond j}} - y_{01{\Diamond j}}} \right)}{\sum\limits_{\iota = 1}^{L_{t}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} - x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} - x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} +} \right.}} \\{\left. {{+ {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} + x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} + x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} - y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} - y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +} \right.}} \\{\left. {{+ {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} + y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} + y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times}\end{matrix} & (0.12) \\\begin{matrix}{{~~~~~~~~~~}{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {z - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z + z_{01{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} -} \right.}} \\{{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z - z_{02{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z + z_{02{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \mspace{11mu} d\; \tau} +}\end{matrix} & \;\end{matrix}$ $\begin{matrix}{\mspace{25mu} {{+ \frac{a}{\left( {\varphi c}_{t} \right)_{j}\left( {d_{j + 1} - d_{j}} \right)\left( {x_{02{\Diamond j}} - x_{01{\Diamond j}}} \right)\left( {y_{02{\Diamond j}} - y_{01{\Diamond j}}} \right)}}{\sum\limits_{\iota = {L_{t} + 1}}^{M_{t}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} - y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} - y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +} \right.}} \\{\left. {{+ {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} + y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} + y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {z - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} + {\Theta_{3}\left\{ {\frac{\pi \left( {z + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \times}}\end{matrix}{\quad {\begin{matrix}{{~~~~~~~~}{\times \left\{ {{\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} -} \right.}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} -}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} - x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} - x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} +}} \\{\left. {{+ {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} + x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} + x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} +}\end{matrix}\quad}}$ $\begin{matrix}{\mspace{25mu} {{+ \frac{b}{\left( {\varphi c}_{t} \right)_{j}\left( {d_{j + 1} - d_{j}} \right)\left( {x_{02{\Diamond j}} - x_{01{\Diamond j}}} \right)\left( {y_{02{\Diamond j}} - y_{01{\Diamond j}}} \right)}}{\sum\limits_{\iota = {M_{\iota} + 1}}^{N_{t}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} - x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} - x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} +} \right.}} \\{\left. {{+ {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} + x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} + x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {z - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {z + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \times}}\end{matrix}{\quad \begin{matrix}{{~~~~~~~~~~}{\times \left\{ {{\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} - y_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} - y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} -} \right.}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} + y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} + y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} -}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} - y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} - y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +}} \\{{\left. {{+ {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} + y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} + y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \mspace{11mu} d\; \tau} +}\end{matrix}}$ $\begin{matrix}{\mspace{20mu} {{+ \frac{1}{2{\pi^{2}\left( {\varphi c}_{t} \right)}_{j}\left( {d_{j + 1} - d_{j}} \right)\left( {x_{02{\Diamond j}} - x_{01{\Diamond j}}} \right)\left( {y_{02{\Diamond j}} - y_{01{\Diamond j}}} \right)}}{\sum\limits_{\iota = {N_{t} + 1}}^{N_{d}}{{U\left( {t - t_{0{\iota j}}} \right)}\mspace{11mu} \sin \mspace{11mu} \vartheta_{0\iota \; j} \times}}}} \\{{\times {\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)}{\int_{z_{01\iota \; j}}^{z_{02{\iota j}}}\left\lbrack \left\{ {{\Theta_{3}^{f}\left( {{\frac{\pi}{2b}\left\{ {y_{02{\Diamond j}} - {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{0\iota \; j}\mspace{11mu} \sin \mspace{11mu} \theta_{0\iota \; j}}} \right\}},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{y\; j}\tau}} \right)} -} \right. \right.}}}}} \\{{{- {\Theta_{3}^{f}\left( {{\frac{\pi}{2b}\left\{ {y_{01{\Diamond j}} - {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{0\iota \; j}\mspace{11mu} \sin \mspace{11mu} \theta_{0\iota \; j}}} \right\}},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{y\; j}\tau}} \right)}} +}} \\{{{+ {\Theta_{3}^{f}\left( {{\frac{\pi}{2b}\left\{ {y_{02{\Diamond j}} + {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{0\iota \; j}\mspace{11mu} \sin \mspace{11mu} \theta_{0\iota \; j}}} \right\}},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{y\; j}\tau}} \right)}} -}} \\{\left. {- {\Theta_{3}^{f}\left( {{\frac{\pi}{2b}\left\{ {y_{01{\Diamond j}} + {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{0\iota \; j}\mspace{11mu} \sin \mspace{11mu} \theta_{0\iota \; j}}} \right\}},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{y\; j}\tau}} \right)}} \right\} \times}\end{matrix}{\quad {\begin{matrix}{{~~~~~~~~}{\times \left\{ {{\Theta_{3}^{f}\left( {{\frac{\pi}{2a}\left\{ {x_{02{\Diamond j}} - {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{0\iota \; j}\mspace{11mu} \sin \mspace{11mu} \theta_{0\iota \; j}}} \right\}},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{x\; j}\tau}} \right)} -} \right.}} \\{{{- {\Theta_{3}^{f}\left( {{\frac{\pi}{2a}\left\{ {x_{01{\Diamond j}} - {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{0\iota \; j}\mspace{11mu} \sin \mspace{11mu} \theta_{0\iota \; j}}} \right\}},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{x\; j}\tau}} \right)}} +}} \\{{{+ {\Theta_{3}^{f}\left( {{\frac{\pi}{2a}\left\{ {x_{02{\Diamond j}} + {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{0\iota \; j}\mspace{11mu} \sin \mspace{11mu} \theta_{0\iota \; j}}} \right\}},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{x\; j}\tau}} \right)}} -}} \\{\left. {- {\Theta_{3}^{f}\left( {{\frac{\pi}{2a}\left\{ {x_{01{\Diamond j}} + {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{0\iota \; j}\mspace{11mu} \sin \mspace{11mu} \theta_{0\iota \; j}}} \right\}},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{x\; j}\tau}} \right)}} \right\} \times} \\{{\left. {\times \left\{ {{\Theta_{3}\left( {{\frac{\pi}{2d}\left( {z - z_{0{\iota j}}} \right)},e^{{- {(\frac{\pi}{d})}^{2}}\eta_{y\; j}\tau}} \right)} + {\Theta_{3}\left( {{\frac{\pi}{2d}\left( {z + z_{0{\iota j}}} \right)},e^{{- {(\frac{\pi}{d})}^{2}}\eta_{y\; j}\tau}} \right)}} \right\}} \right\rbrack \mspace{11mu} {dz}_{0\iota \; j}d\; \tau} +}\end{matrix}\quad}}$ $\begin{matrix}{\mspace{31mu} {{+ \frac{2{ab}}{\left( {\varphi c}_{t} \right)_{j}\left( {d_{j + 1} - d_{j}} \right)\left( {x_{02{\Diamond j}} - x_{01{\Diamond j}}} \right)\left( {y_{02{\Diamond j}} - y_{01{\Diamond j}}} \right)}}{\sum\limits_{\iota = {N_{d} + 1}}^{L_{r}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} -} \right.}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} -}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} - x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} - x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} +}} \\{\left. {{+ {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} + x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} + x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} +}\end{matrix}{\quad {\begin{matrix}{{~~~~~~~~~~}{\times \left\{ {{\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} - y_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} - y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} -} \right.}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} + y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} + y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} -}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} - y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} - y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +}} \\{\left. {{+ {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} + y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} + y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times} \\{{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {z - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {z + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \mspace{11mu} d\; \tau} +}}\end{matrix}\quad}}$ $\begin{matrix}{\mspace{25mu} {{+ \frac{2b}{\left( {\varphi c}_{t} \right)_{j}\left( {x_{02{\Diamond j}} - x_{01{\Diamond j}}} \right)\left( {y_{02{\Diamond j}} - y_{01{\Diamond j}}} \right)}}{\sum\limits_{\iota = {L_{r} + 1}}^{M_{r}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} - x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} - x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} +} \right.}} \\{\left. {{+ {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} + x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} + x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} - y_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} - y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} -} \right.}}\end{matrix}{\quad {\begin{matrix}{{~~~~~~~~~}{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} + y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} + y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} -}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} - y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} - y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +}} \\{\left. {{+ {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} + y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} + y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {z - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z + z_{01{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} -} \right.}} \\{{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z - z_{02{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z + z_{02{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \mspace{11mu} d\; \tau} +}\end{matrix}\quad}}$ $\begin{matrix}{\mspace{25mu} {{+ \frac{2a}{\left( {\varphi c}_{t} \right)_{j}\left( {x_{02{\Diamond j}} - x_{01{\Diamond j}}} \right)\left( {y_{02{\Diamond j}} - y_{01{\Diamond j}}} \right)}}{\sum\limits_{\iota = {M_{r} + 1}}^{N_{r}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} -} \right.}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} -}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} - x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} - x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} +}} \\{\left. {{+ {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{02{\Diamond j}} + x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {x_{01{\Diamond j}} + x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} +}\end{matrix}{\quad \begin{matrix}{{~~~~~~~~~~}{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} - y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} - y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +} \right.}} \\{\left. {{+ {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} + y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} + y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {z - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z + z_{01{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} -} \right.}} \\{{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z - z_{02{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z + z_{02{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \mspace{11mu} d\; \tau} +}\end{matrix}}$ $\quad{\begin{matrix}{{~~~~~~~~~}{{+ \frac{1}{\left( {\varphi c}_{t} \right)_{j}\left( {d_{j + 1} - d_{j}} \right)\left( {x_{02{\Diamond j}} - x_{01{\Diamond j}}} \right)\left( {y_{02{\Diamond j}} - y_{01{\Diamond j}}} \right)}} \times}} \\{{\times {\int_{0}^{t}{\int_{0}^{a}{\int_{0}^{b}\left\lbrack {{{\psi_{j}\left( {u,v,\tau} \right)}\Theta_{3}\left\{ {\frac{\pi \left( {z - d_{j}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}({t - \tau})}}} \right\}} -} \right.}}}}} \\{\left. {{- {\psi_{j + 1}\left( {u,v,\tau} \right)}}\Theta_{4}\left\{ {\frac{\pi \left( {z - d_{j}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}({t - \tau})}}} \right\}} \right\rbrack \times}\end{matrix}\begin{matrix}{{~~~~~~~~~~}{\times \left\lbrack {{\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {x_{02{\Diamond j}} - u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {x_{01{\Diamond j}} - u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}} +} \right.}} \\{\left. {{{+ \Theta_{3}^{\int}}\left\{ {\frac{\pi \left( {x_{02{\Diamond j}} + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {x_{01{\Diamond j}} + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}}} \right\rbrack \times} \\{{\times \left\lbrack {{\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {y_{02{\Diamond j}} - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {y_{01{\Diamond j}} - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} +} \right.}} \\{{\left. {{{+ \Theta_{3}^{\int}}\left\{ {\frac{\pi \left( {y_{02{\Diamond j}} + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {y_{01{\Diamond j}} + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}}} \right\rbrack \mspace{11mu} {dudvd}\; \tau} +}\end{matrix}}$ $\begin{matrix}{\mspace{25mu} {{+ \frac{1}{\left( {\varphi c}_{t} \right)_{j}\left( {d_{j + 1} - d_{j}} \right)\left( {x_{02{\Diamond j}} - x_{01{\Diamond j}}} \right)\left( {y_{02{\Diamond j}} - y_{01{\Diamond j}}} \right)}} \times}} \\{{\times {\int_{0}^{t}{\int_{0}^{b}{\int_{0}^{d_{j + 1} - d_{j}}\left\{ {{{\psi_{0{yzj}}\left( {v,w,\tau} \right)}\mspace{11mu} \left\{ {{\Theta_{3}^{\int}\left( {\frac{{\pi x}_{02{\Diamond j}}}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{zj}({t - \tau})}}} \right)} - {\Theta_{3}^{\int}\left( {\frac{{\pi x}_{01{\Diamond j}}}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{zj}({t - \tau})}}} \right)}} \right\}} -} \right.}}}}} \\{\left. {{- {\psi_{ayzj}\left( {v,w,\tau} \right)}}\left\{ {{\Theta_{4}^{\int}\left( {\frac{{\pi x}_{02{\Diamond j}}}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}({t - \tau})}}} \right)} - {\Theta_{4}^{\int}\left( {\frac{{\pi x}_{01{\Diamond j}}}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}({t - \tau})}}} \right)}} \right\}} \right\} \times} \\{{\times \left\lbrack {{\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {x_{02{\Diamond j}} - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {y_{01{\Diamond j}} - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} +} \right.}} \\{\left. {{{+ \Theta_{3}^{\int}}\left\{ {\frac{\pi \left( {y_{02{\Diamond j}} + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {y_{01{\Diamond j}} + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}}} \right\rbrack \times} \\{{{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {z - d_{j\;} - w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {z - d_{j\;} + w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}}} \right\rbrack \; {dvdwd}\; \tau} +}}\end{matrix}\quad$ $\begin{matrix}{\mspace{20mu} {{+ \frac{1}{\left( {\varphi c}_{t} \right)_{j}\left( {d_{j + 1} - d_{j}} \right)\left( {x_{02{\Diamond j}} - x_{01{\Diamond j}}} \right)\left( {y_{02{\Diamond j}} - y_{01{\Diamond j}}} \right)}} \times}} \\{{\times {\int_{0}^{t}{\int_{0}^{a}{\int_{0}^{d_{j + 1} - d_{j}}\left\{ {{{\psi_{x\; 0{zj}}\left( {u,w,\tau} \right)}\mspace{11mu} \left\{ {{\Theta_{3}^{\int}\left( {\frac{{\pi y}_{02{\Diamond j}}}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}({t - \tau})}}} \right)} - {\Theta_{3}^{\int}\left( {\frac{{\pi y}_{01{\Diamond j}}}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}({t - \tau})}}} \right)}} \right\}} -} \right.}}}}} \\{\left. {{- {\psi_{xbzj}\left( {u,w,\tau} \right)}}\left\{ {{\Theta_{4}^{\int}\left( {\frac{{\pi y}_{02{\Diamond j}}}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}({t - \tau})}}} \right)} - {\Theta_{4}^{\int}\left( {\frac{{\pi y}_{01{\Diamond j}}}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}({t - \tau})}}} \right)}} \right\}} \right\} \times} \\{{\times \left\lbrack {{\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {x_{02{\Diamond j}} - u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {x_{01{\Diamond j}} - u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} +} \right.}} \\{\left. {{{+ \Theta_{3}^{\int}}\left\{ {\frac{\pi \left( {x_{02{\Diamond j}} + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {x_{01{\Diamond j}} + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}}} \right\rbrack \times} \\{{{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {z - d_{j\;} - w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {z - d_{j\;} + w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}}} \right\rbrack \; {dudwd}\; \tau} +}}\end{matrix}\quad$ $\begin{matrix}{\mspace{25mu} {{+ \frac{1}{2\left( {d_{j + 1} - d_{j}} \right)\left( {x_{02{\Diamond j}} - x_{01{\Diamond j}}} \right)\left( {y_{02{\Diamond j}} - y_{01{\Diamond j}}} \right)}} \times}} \\{{\times {\int_{0}^{d_{j + 1} - d_{j}}{\int_{0}^{b}{\int_{0}^{a}{{\phi_{j}\left( {u,v,{w + d_{j}}} \right)}\mspace{11mu}\left\lbrack {{\Theta_{3}^{\int}\left\{ {\frac{\pi\left( {x_{02{\Diamond j}} + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{zj}({t - \tau})}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {x_{01{\Diamond j}} + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{zj}({t - \tau})}}} \right\}} +} \right.}}}}}} \\{\left. {{{+ \Theta_{3}^{\int}}\left\{ {\frac{\pi\left( {x_{02{\Diamond j}} + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}({t - \tau})}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {x_{01{\Diamond j}} + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}({t - \tau})}}} \right\}}} \right\rbrack \times} \\{{\times \left\lbrack {{\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {y_{02{\Diamond j}} - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {y_{01{\Diamond j}} - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} +} \right.}} \\{\left. {{{+ \Theta_{3}^{\int}}\left\{ {\frac{\pi \left( {y_{02{\Diamond j}} + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {y_{01{\Diamond j}} + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}}} \right\rbrack \times} \\{\left. {\times \left\{ {{\Theta_{3}\left\{ {\frac{\pi \left( {z - d_{j\;} - w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}t}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {z - d_{j\;} + w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}t}} \right\}}} \right\}} \right\rbrack \; {dudvdw}}\end{matrix}$

TABLE 7The  spatial  average  pressure  response  of  the  rectangle  [(y_(02◇j) − y_(01◇j))(z_(02◇j) − z_(01◇j))], ι = ◇, L_(r) + 1 ≤ ◇ ≤ M_(r)  is  given  by$\begin{matrix}\begin{matrix}{p_{j} = {\frac{\; \left( {d_{j + 1} - d_{j}} \right)}{\left( {\varphi c}_{t} \right)_{j}{a\left( {y_{02{\Diamond j}} - y_{01{\Diamond j}}} \right)}\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}{\sum\limits_{\iota = 1}^{L_{t}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {x - x_{0j}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{zj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {x + x_{0j}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} - y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} - y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +} \right.}} \\{\left. {{+ {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} + y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} + y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times}\end{matrix} & (0.13) \\{\mspace{59mu} \begin{matrix}{{\times \left\{ {{\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{01{\Diamond j}} - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} -} \right.}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} + z_{01{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{01{\Diamond j}} + z_{01{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} -}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} - z_{02{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} +}} \\{{\left. {{+ {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} + z_{02{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{01{\Diamond j}} + z_{02{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \mspace{11mu} {d\tau}} +}\end{matrix}} & \;\end{matrix}$ $\begin{matrix}{\mspace{40mu} {{+ \frac{1}{\left( {\varphi c}_{t} \right)_{j}\left( {y_{02{\Diamond j}} - y_{01{\Diamond j}}} \right)\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}}{\sum\limits_{\iota = {L_{t} + 1}}^{M_{t}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} - y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} - y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +} \right.}} \\{\left. {{+ {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} + y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} + y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times}\end{matrix}$ $\begin{matrix}{{~~~~~~~~~~~~~}{\times \left\{ {{\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{02{\Diamond j}} - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}({t - \tau})}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} +} \right.}} \\{\left. {{{+ \Theta_{3}^{\int}}\left\{ {\frac{\pi \left( {z_{02{\Diamond j}} + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}({t - \tau})}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}({t - \tau})}}} \right\}}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xy}\tau}} \right)} -} \right.}} \\{{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} d\; \tau} +}\end{matrix}\quad$ $\begin{matrix}{\mspace{40mu} {{+ \frac{b}{\left( {\varphi c}_{t} \right)_{j}{a\left( {y_{02{\Diamond j}} - y_{01{\Diamond j}}} \right)}\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}}{\sum\limits_{\iota = {M_{\iota} + 1}}^{N_{t}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {x - x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {x + x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{02{\Diamond j}} - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}({t - \tau})}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} +} \right.}} \\{\left. {{{+ \Theta_{3}^{\int}}\left\{ {\frac{\pi \left( {z_{02{\Diamond j}} + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}({t - \tau})}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}({t - \tau})}}} \right\}}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} - y_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} - y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} -} \right.}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} + y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} + y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} -}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} - y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} - y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +}} \\{{\left. {{+ {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} + y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} + y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \; d\; \tau} +}\end{matrix}\quad$ $\begin{matrix}{\mspace{31mu} {{+ \frac{1}{2{\pi^{2}\left( {\varphi c}_{t} \right)}_{j}a\left( {y_{02{\Diamond j}} - y_{01{\Diamond j}}} \right)\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}}\; {\sum\limits_{\iota = {N_{l} + 1}}^{N_{d}}{{U\left( {t - t_{0{\iota j}}} \right)}\mspace{11mu} \sin \mspace{11mu} \vartheta_{0\iota \; j} \times}}}} \\{{\times {\int_{0}^{t - t_{0{\iota j}}}{q_{j}\; \left( {t - t_{0{\iota j}} - \tau} \right){\int_{z\; 01{\iota j}}^{z_{02{\iota j}}}\left\lbrack \left\{ {{\Theta_{3}\left( {{\frac{\pi}{2a}\left\{ {x - {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{0{\iota j}}\cos \mspace{11mu} \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{a})}^{2}}{\eta {xj}\tau}}} \right)} +} \right. \right.}}}}} \\{\left. {+ {\Theta_{3}\left( {{\frac{\pi}{2a}\left\{ {x + {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{{0{\iota j}}\;}\cos \mspace{11mu} \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{a})}^{2}}{\eta {xj}\tau}}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {{\frac{\pi}{2b}\left\{ {y_{02{\Diamond j}} - {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{0{\iota j}}\mspace{11mu} \sin \mspace{11mu} \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{b})}^{2}}\eta \; {yj\tau}}} \right)} -} \right.}} \\{{{- {\Theta_{3}^{\int}\left( {{\frac{\pi}{2b}\left\{ {y_{01{\Diamond j}} - {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{0{\iota j}}\mspace{11mu} \sin \mspace{11mu} \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{b})}^{2}}\eta \; {yj\tau}}} \right)}} +}} \\{{{+ {\Theta_{3}^{\int}\left( {{\frac{\pi}{2b}\left\{ {y_{02{\Diamond j}} + {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{0{\iota j}}\mspace{11mu} \sin \mspace{11mu} \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{b})}^{2}}\eta \; {yj\tau}}} \right)}} -}} \\{\left. {- {\Theta_{3}^{\int}\left( {{\frac{\pi}{2b}\left\{ {y_{01{\Diamond j}} + {\left( {z_{0{\iota j}} - \gamma_{0{\iota j}}} \right)\mspace{11mu} \cot \mspace{11mu} \vartheta_{0{\iota j}}\mspace{11mu} \sin \mspace{11mu} \theta_{0{\iota j}}}} \right\}},e^{{- {(\frac{\pi}{b})}^{2}}\eta \; {yj\tau}}} \right)}} \right\} \times}\end{matrix}{\quad \begin{matrix}{{~~~~~~~~~~~}{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {\eta_{zj}{(\frac{\pi}{d_{j + 1} - d_{j}})}}^{2}}t}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z_{01{\Diamond j}} - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {\eta_{zj}{(\frac{\pi}{d_{j + 1} - d_{j}})}}^{2}}t}} \right)} +} \right.}} \\{{\left. \left. {{+ {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {\eta_{zj}{(\frac{\pi}{d_{j + 1} - d_{j}})}}^{2}}t}} \right)}} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {z_{01{\Diamond j}} + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {\eta_{zj}{(\frac{\pi}{d_{j + 1} - d_{j}})}}^{2}}t}} \right)}} \right\} \right\rbrack \mspace{11mu} {dz}_{0\iota \; j}d\; \tau} +}\end{matrix}}$ $\begin{matrix}{\mspace{40mu} {{+ \frac{2b}{\left( {\varphi c}_{t} \right)_{j}\left( {y_{02{\Diamond j}} - y_{01{\Diamond j}}} \right)\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}}{\sum\limits_{\iota = {N_{d} + 1}}^{L_{r}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{zj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{zj}\tau}} \right)} -} \right.}} \\{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} - y_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} - y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} -} \right.}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} + y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} + y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} -}}\end{matrix}{\quad \mspace{59mu} \begin{matrix}{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} - y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} - y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +}} \\{\left. {{+ {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} + y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} + y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times} \\{{{\times \left\{ {{\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{02{\Diamond j}} - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}({t - \tau})}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} - z_{0{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right)}} \right\}} +}} \\{{\left. {{{+ \Theta_{3}^{\int}}\left\{ {\frac{\pi \left( {z_{02{\Diamond j}} + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} + z_{0{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}}} \right\} \mspace{11mu} {d\tau}} +}\end{matrix}}$ $\begin{matrix}{\mspace{40mu} {{+ \frac{2b\; \left( {d_{j + 1} - d_{j}} \right)}{\left( {\varphi c}_{t} \right)_{j}{a\left( {y_{02{\Diamond j}} - y_{01{\Diamond j}}} \right)}\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}}{\sum\limits_{\iota = {L_{r} + 1}}^{M_{r}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}\left( {\frac{\pi \left( {x - x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {x + x_{0{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} \times}} \\{{\times \left\{ {{\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} - y_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} - y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} -} \right.}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} + y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} + y_{01{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} -}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} - y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} - y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +}} \\{\left. {{+ {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} + y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} + y_{02{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times}\end{matrix}{\quad \mspace{59mu} {\begin{matrix}{{\times \left\{ {{\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{01{\Diamond j}} - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} -} \right.}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} + z_{01{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{01{\Diamond j}} + z_{01{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} -}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} - z_{02{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} +}} \\{{\left. {{+ {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} + z_{02{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{01{\Diamond j}} + z_{02{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \mspace{11mu} {d\tau}} +}\end{matrix}\quad}}$ $\begin{matrix}{\mspace{40mu} {{+ \frac{2\; \left( {d_{j + 1} - d_{j}} \right)}{\left( {\varphi c}_{t} \right)_{j}\left( {y_{02{\Diamond j}} - y_{01{\Diamond j}}} \right)\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}}{\sum\limits_{\iota = {M_{r} + 1}}^{N_{r}}{{U\left( {t - t_{0{\iota j}}} \right)}{\int_{0}^{t - t_{0{\iota j}}}{{q_{\iota j}\left( {t - t_{0{\iota j}} - \tau} \right)} \times}}}}}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{01{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)} -} \right.}} \\{\left. {{- {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x - x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} + {\Theta_{3}^{\int}\left( {\frac{\pi \left( {x + x_{02{\iota j}}} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}\tau}} \right)}} \right\} \times} \\{{\times \left\{ {{\Theta_{3}^{\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} - y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} - y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)} +} \right.}} \\{\left. {{+ {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y_{02{\Diamond j}} + y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} - {\Theta_{3}^{\int}\left( {\frac{\pi \left( {y_{01{\Diamond j}} + y_{0{\iota j}}} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}\eta_{yj}\tau}} \right)}} \right\} \times}\end{matrix}{\quad \mspace{59mu} {\begin{matrix}{{\times \left\{ {{\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{01{\Diamond j}} - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} -} \right.}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} + z_{01{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{01{\Diamond j}} + z_{01{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} -}} \\{{{- {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} - z_{02{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} + {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} - z_{01{\iota j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)} +}} \\{{\left. {{\times {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{02{\Diamond j}} + z_{02{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} - {\Theta_{3}^{\int\;\int}\left( {\frac{\pi \left( {z_{01{\Diamond j}} + z_{02{\iota j}} - {2d_{j}}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}\tau}} \right)}} \right\} \mspace{11mu} {d\tau}} +}\end{matrix}\quad}}$ $\begin{matrix}{\mspace{40mu} {{+ \frac{1}{\left( {\varphi c}_{t} \right)_{j}{a\left( {y_{02{\Diamond j}} - y_{01{\Diamond j}}} \right)}\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}} \times}} \\{{\times {\int_{0}^{t}{\int_{0}^{a}{\int_{0}^{b}\left\{ {{\psi_{j}\left( {u,v,\tau} \right)}\left\{ {{\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{02{\iota j}} - d_{j}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}({t - \tau})}}} \right\}} -} \right.} \right.}}}}} \\{\left. {{- \Theta_{3}^{\int}}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} - d_{j}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} \right\} -} \\{{{- {\psi_{j + 1}\left( {u,v,\tau} \right)}}\left\{ {{\Theta_{4}^{\int}\left\{ {\frac{\pi \left( {z_{02{\Diamond j}} - d_{j}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}({t - \tau})}}} \right\}} -} \right.}} \\{\left. \left. {{- \Theta_{4}^{\int}}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} - d_{j}} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} \right\} \right\rbrack \times}\end{matrix}{\quad {\begin{matrix}{{~~~~~~~~~~~~~}{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {x - u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {x + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}}} \right\rbrack \times}} \\{{\times \left\lbrack {{\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {y_{02{\Diamond j}} - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {y_{01{\Diamond j}} - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} +} \right.}} \\{{\left. {{{+ \Theta_{3}^{\int}}\left\{ {\frac{\pi \left( {y_{02{\Diamond j}} + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {y_{01{\Diamond j}} + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}}} \right\rbrack \mspace{11mu} {dudvd}\; \tau} +}\end{matrix}\quad}}$ $\begin{matrix}{\mspace{40mu} {{+ \frac{1}{\left( {\varphi c}_{t} \right)_{j}{a\left( {y_{02{\Diamond j}} - y_{01{\Diamond j}}} \right)}\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}} \times}} \\{{\times {\int_{0}^{t}{\int_{0}^{b}{\int_{0}^{d_{j + 1} - d_{j}}{\left\{ {{{\psi_{0{yzj}}\left( {v,w,\tau} \right)}\mspace{11mu} {\Theta_{3}\left( {\frac{\pi x}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}({t - \tau})}}} \right)}} - {{\psi_{ayzj}\left( {v,w,\tau} \right)}\mspace{11mu} {\Theta_{4}\left( {\frac{\pi x}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}({t - \tau})}}} \right)}}} \right\} \times}}}}}} \\{{\times \left\lbrack {{\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {y_{02{\Diamond j}} - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {y_{01{\Diamond j}} - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} +} \right.}} \\{\left. {{{+ \Theta_{3}^{\int}}\left\{ {\frac{\pi \left( {y_{02{\Diamond j}} + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {y_{01{\Diamond j}} + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}}} \right\rbrack \times} \\{{\times \left\{ {{\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{02{\Diamond j}} - d_{j} - w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}({t - \tau})}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} - d_{j} - w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} +} \right.}} \\{{\left. {{\times \Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{02{\Diamond j}} - d_{j} + w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} - d_{j} + w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}}} \right\} {dvdwd\tau}} +}\end{matrix}\quad$ $\begin{matrix}{\mspace{40mu} {{+ \frac{1}{\left( {\varphi c}_{t} \right)_{j}{a\left( {y_{02{\Diamond j}} - y_{01{\Diamond j}}} \right)}\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}} \times}} \\{{\times {\int_{0}^{t}{\int_{0}^{a}{\int_{0}^{d_{j + 1} - d_{j}}\left\{ {{{\psi_{x0zj}\left( {v,w,\tau} \right)}{\; \;}\left\{ {{\Theta_{3}^{\int}\left( {\frac{{\pi y}_{02{\Diamond j}}}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right)} - {\Theta_{3}^{\int}\left( {\frac{{\pi y}_{01{\Diamond j}}}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right)}} \right\}} -} \right.}}}}} \\{\left. {{- {\psi_{xbzj}\left( {v,w,\tau} \right)}}\left\{ {{\Theta_{4}^{\int}\left( {\frac{{\pi y}_{02{\Diamond j}}}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}({t - \tau})}}} \right)} - {\Theta_{4}^{\int}\left( {\frac{{\pi y}_{01{\Diamond j}}}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}({t - \tau})}}} \right)}} \right\}} \right\} \times} \\{{\times \left\lbrack {{\Theta_{3}\left\{ {\frac{\pi \left( {x - u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}} + {\Theta_{3}\left\{ {\frac{\pi \left( {x + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}{\eta_{xj}{({t - \tau})}}}} \right\}}} \right\rbrack \times}} \\{{\times \left\{ {{\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{02{\Diamond j}} - d_{j} - w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}({t - \tau})}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} - d_{j} - w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} +} \right.}} \\{{\left. {{\times \Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{02{\Diamond j}} - d_{j} + w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} - d_{j} + w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}{\eta_{zj}{({t - \tau})}}}} \right\}}} \right\} \mspace{11mu} {dvdwd}\; \tau} +}\end{matrix}\quad$ $\begin{matrix}{\mspace{40mu} {{+ \frac{1}{2a\; \left( {y_{02{\Diamond j}} - y_{01{\Diamond j}}} \right)\left( {z_{02{\Diamond j}} - z_{01{\Diamond j}}} \right)}} \times}} \\{{\times {\int_{0}^{d_{j + 1} - d_{j}}{\int_{0}^{b}{\int_{0}^{a}{{\phi_{j}\left( {u,v,{w + d_{j}}} \right)}\left\lbrack {\left\{ {{\Theta_{3}\left( {\frac{\pi \left( {x - u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}t}} \right)} + {\Theta_{3}\left( {\frac{\pi \left( {x + u} \right)}{2a},e^{{- {(\frac{\pi}{a})}^{2}}\eta_{xj}t}} \right)}} \right\} \times} \right.}}}}}} \\{{\times \left\lbrack {{\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {y_{02{\Diamond j}} - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {y_{01{\Diamond j}} - v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} +} \right.}} \\{\left. {{{+ \Theta_{3}^{\int}}\left\{ {\frac{\pi \left( {y_{02{\Diamond j}} + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {y_{01{\Diamond j}} + v} \right)}{2b},e^{{- {(\frac{\pi}{b})}^{2}}{\eta_{yj}{({t - \tau})}}}} \right\}}} \right\rbrack \times} \\{{\times \left\{ {{\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{02{\Diamond j}} - d_{j} - w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}t}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} - d_{j} - w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}t}} \right\}} +} \right.}} \\{\left. {{\times \Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{02{\Diamond j}} - d_{j} + w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}t}} \right\}} - {\Theta_{3}^{\int}\left\{ {\frac{\pi \left( {z_{01{\Diamond j}} - d_{j} + w} \right)}{2\left( {d_{j + 1} - d_{j}} \right)},e^{{- {(\frac{\pi}{d_{j + 1} - d_{j}})}^{2}}\eta_{zj}t}} \right\}}} \right\} {dudvdw}}\end{matrix}\quad$

The nomenclatures used in TABLES 3 through 7 are listed in TABLE 8below.

TABLE 8 Nomenclature a Width of the layer, m. b Bredth of the layer, m.c_(t)  Compressibility, P_(a)⁻¹. φ Porosity, fraction. d_(j+1) − d_(j)Layer thickness, m. k_(x), k_(y), k_(z) Permeability in the x, y and zdirection, m² μ Viscosity, P_(a) · s.${\eta_{xj} = \left( \frac{k_{x}}{{\varphi c}_{t}\mu} \right)_{j}},{\eta_{yj} = {{\left( \frac{k_{y}}{{\varphi c}_{t}\mu} \right)_{j}\mspace{14mu} {and}\mspace{20mu} \eta_{zj}} = {\left( \frac{k_{z}}{{\varphi c}_{t}\mu} \right)_{j}\mspace{14mu} {Diffusion}\mspace{14mu} {coefficients}}}}$p_(j) Pressure in the jth layer, P_(a). q_(ιj) Production rate of theιth well or fracture in the jth layer, m³/s. t Time, s. t_(0ιj) Stattime of production of the ιth well or fracture in the jth layer, s.θ_(0ιj) The inclination to the x − y plane of the ιth well or fracturein the jth layer γ_(0j) The intercept to the z axis of the ιth well orfracture in the jth layer${U\left( {t - t_{0}} \right)} = \left\{ {{{\begin{matrix}0 & {t < t_{0}} \\1 & {t > t_{0}}\end{matrix}{Heaviside}\text{'}s\mspace{14mu} {Unit}\mspace{14mu} {step}\mspace{14mu} {function}\mspace{14mu} s\mspace{14mu} {Laplace}\mspace{14mu} {{variable}.}} \ni_{m}} = \left\{ \begin{matrix}\frac{1}{2} & {m = 0} \\1 & {{m = 1},2,3,\ldots}\end{matrix} \right.} \right.$ $\begin{matrix}{{\Theta_{3}\left( {{\pi \; x},e^{{- \pi^{2}}t}} \right)} = {\begin{Bmatrix}{1 + {2\underset{n = 1}{\overset{\infty}{\;\sum}}e^{{- n^{2}}\pi^{2}t}\cos \; \left( {2n\; {\pi x}} \right)}} & {e^{{- \pi^{2}}t} > \frac{1}{\pi}} \\{\frac{1}{\sqrt{\pi t}}{\sum\limits_{n = {- \infty}}^{\infty}e^{- \frac{{({x + n})}^{2}}{t}}}} & {e^{{- \pi^{2}}t} \leq \frac{1}{\pi}}\end{Bmatrix}\mspace{11mu} {Eliptic}\mspace{14mu} {theta}\mspace{14mu} {function}\mspace{14mu} {of}\mspace{14mu} {the}\mspace{14mu} {third}\mspace{14mu} {kind}}} \\{{\Theta_{3}^{\int}\left( {{\pi \; x},e^{{- \pi^{2}}t}} \right)} = {{\int_{0}^{x}{{\Theta_{3}\left( {{\pi u},e^{{- \pi^{2}}t}} \right)}\; {du}}} = {\begin{Bmatrix}{x + {\frac{1}{\pi}\underset{n = 1}{\overset{\infty}{\;\sum}}\frac{e^{{- n^{2}}\pi^{2}t}}{n}\sin \; \left( {2n\; {\pi x}} \right)}} & {e^{{- \pi^{2}}t} > \frac{1}{\pi}} \\{\frac{1}{2}{\sum\limits_{n = {- \infty}}^{\infty}\left\{ {{{erf}\mspace{11mu} \left( \frac{x + n}{\sqrt{t}} \right)} - {{erf}\mspace{14mu} \left( \frac{n}{\sqrt{t}} \right)}} \right\}}} & {e^{{- \pi^{2}}t} \leq \frac{1}{\pi}}\end{Bmatrix}\mspace{11mu} {Integral}\mspace{14mu} {of}\mspace{14mu} {Eliptic}}}}\end{matrix}$ theta function of the third kind $\begin{matrix}{{\Theta_{3}^{\int\;\int}\left( {{\pi x},e^{{- \pi^{2}}t}} \right)} = {{\int_{0}^{x}{{\Theta_{3}^{\int}\left( {{\pi u},e^{{- \pi^{2}}t}} \right)}\; {du}}} =}} \\{= \begin{Bmatrix}{\frac{x^{2}}{2} + {\frac{1}{2\pi^{2}}{\sum\limits_{n = 1}^{\infty}{\frac{e^{{- n^{2}}\pi^{2}t}}{n^{2}}\left\{ {1 - {\cos \; \left( {2n\; {\pi x}} \right)}} \right\}}}}} & {e^{{- \pi^{2}}t} > \frac{1}{\pi}} \\{\frac{1}{2}{\sum\limits_{n = {- \infty}}^{\infty}\left\{ {{\left( {x + n} \right)\mspace{11mu} {erf}\mspace{11mu} \left( \frac{x + n}{\sqrt{t}} \right)} + {\sqrt{\frac{t}{\pi}\;}\left( {e^{- \frac{{({x + n})}^{2}}{t}} - e^{- \frac{n^{2}}{t}}} \right)} - {n\mspace{14mu} {erf}\mspace{14mu} \left( \frac{n}{\sqrt{t}} \right)}} \right\}}} & {e^{{- \pi^{2}}t} \leq \frac{1}{\pi}}\end{Bmatrix}}\end{matrix}$ Second integral of Eliptic theta function of the thirdkind

FIG. 9 is a flow chart of a method to perform an oilfield operationusing the real-time analytical simulator. The oilfield operation isperformed in an oilfield, such as the oilfield (300) depicted in FIG. 3above. This method involves using the gridless analytical simulator,such as described with respect to FIG. 8, to generate real-timesimulation results for performing the oilfield operation.

In Step 901, multiple real-time parameters are obtained from sensorsdisposed about the oilfield (e.g., oilfield (300)). The oilfield mayinclude multiple wellsites, such as depicted in FIG. 3 above. Themultiple real-time parameters include, at least, real-time flow ratedata, real-time pressure data, or real-time temperature data of thewellbore (e.g., wellbore (436) of FIG. 5). These real-time data may bemonitored by a user (e.g., the surveillance engineer). In some examples,there may be missing periods of the real-time measurement, which may besupplemented with data re-construction, for example based on tubing heador bottom hole pressure measurement (Step 902). The real-time pressuredata and/or the real-time flow rate data (if available) are filtered,for example by using a wavelet decomposition technique to removeoutliers, noise, and identify transients (Step 902). The large amount ofreal-time raw data may be sampled to reduce to the filtered data to amanageable amount, while retaining all the relevant characteristics ofthe original larger data set.

A set of alarm conditions is calculated based on the real-time dataafter filtering (Step 903). The alarms may include, for example, adrawdown alarm, a downtime alarm, etc. If the alarm is triggered,detailed diagnostics are performed thereafter. For example, drawdownpressure may be selected as the alarm parameter where the runningmaximum and running minimum values for pressure are calculated for eachhour. These running averages are reset at the end of each hour. Therunning maximum, minimum, and average of the pressure data are alsocalculated for the day. The running averages are reset at 24:00:00 eachday. Static reservoir pressure (Pr) in the vicinity of the well bore isestimated and entered at predefined intervals, typically every 48 to 72hours.

Occasionally, previously estimated Pr values are re-estimated, in whichcase other previously estimated values must be updated. Drawdownpressures are calculated by subtracting the gauge pressure (Pwg) fromthe static reservoir pressure (Pr). Limiting values for gauge pressureare calculated or estimated and entered at predefined intervals,typically 48 to 72 hours.

The sources are bubblepoint limits, sand management limits and drawdownlimits. Bubblepoint limits are absolute limits for the bottomholepressure; sand management limits are functions of the static reservoirpressure; drawdown limits are a fixed offset from the static reservoirpressure. Occasionally, these limits are recomputed, and the previousvalues must be updated.

Drawdown surveillance is performed each hour by comparing the hourlyaverage, running maxima, running minima, and running averages to theappropriate limiting values for gauge pressure. Automatic alerts (e.g.,indicated in color yellow) are generated whenever the gauge pressure iswithin a defined variance from the limit value.

A surveillance engineer analyzes automatic alerts and sets a validationcondition for each alert (e.g., Green: “No action;” Yellow: “Monitorclosely:” Red: “Action recommended”) with an optional comment. Greenmeasurements indicate that a component or system is performing withinspecified bounds and requires no action. Essentially, green-light datacan be ignored. Yellow is a low level alarm (or alert), meaning thesensor measurement is approaching upper or lower bounds. Red is an alarm(or critical level alert), which indicates that the component has beenshut down because sensor measurements fall outside of specified ranges.The yellow alert is one key to asset management, helping operators avoiddeferred production. Operators take proactive measures on yellow alerts,and are reactive to red alarms. Alternatively, other colors may also beused in lieu of the Green/Yellow/Red system.

While the drawdown pressure can be directly calculated from the measuredreal-time data in the above example, wellbore skin may be selected asthe alarm parameter in another example where the running maximum andrunning minimum values for wellbore skin are calculated on a regularbasis using the gridless simulator. Within the gridless analyticalsimulator, many parameters may be used to configure an appropriate modelfor simulating the oilfield (e.g., the oilfield (300)) (Step 904). Forexample, static parameters obtained through geological surveys (e.g. asdepicted in FIG. 1 and FIG. 3 above) may be used to set up the initialand boundary conditions described in TABLE 1 above.

Based on the configurations of the wellsites (e.g., vertical well,horizontal well, deviated well, fractured well, etc.), the gridlessanalytical simulator is configured using equations shown in TABLES 3through 7 above. For example, the coefficients in equation (0.13) areappropriately determined for each well configuration. Preferably, themodel is further identified by using a neural network method based on,for example rate of change of the real-time pressure data. Additionally,a history matching method of key parameters, such as the historic valueof the reservoir pressure, well skin, effective permeability, and wellproductivity may be used to update the model further.

Once the model is identified and the simulator is configured, real-timesimulation results are then generated, for example based on equationsdescribed in TABLES 3-7 above (Step 905). The real-time simulationresults include a prediction of the production rates and reservoirpressure over time. The real-time simulation results can be delivered inan automatic workflow with real-time plotting of the key parameters(e.g., the reservoir pressure, well skin, effective permeability, wellproductivity, etc.) and alarm setting based on pre-determined criteria.The model is automatically updated when the predicted performancediverges from the actual performance by more than a pre-determined limit(Step 906).

In Step 907, the oilfield operation is performed based on the real-timesimulation results. The gridless analytical simulator may provideinformation indicating problems at the wellsites that require action.The simulators may also indicate that adjustments in the oilfieldoperation may be made to improve efficiency, or correct problems. Wellmanagement strategy may be adjusted to define different developmentscenarios to be included in the integrated simulation run.

The steps of portions or all of the process may be repeated as desired.Repeated steps may be selectively performed until satisfactory resultsachieved. For example, steps may be repeated after adjustments are made.This may be done to update the simulator and/or to determine the impactof changes made.

The data input, coupling, layout, and constraints defined in thesimulation provide flexibility to the simulation process. These factorsof the various simulators are selected to meet the requirements of theoilfield operation. Any combination of simulators may be selectivelylinked to create the overall oilfield simulation. The process of linkingthe simulators may be re-arranged and simulations repeated usingdifferent configurations. Depending on the type of coupling and/or thearrangement of simulators, the oilfield simulation may be selected toprovide the desired results. Various combinations may be tried andcompared to determine the best outcome. Adjustments to the oilfieldsimulation may be made based on the oilfield, the simulators, thearrangement, and other factors. The process may be repeated as desired.

It will be understood from the foregoing description that variousmodifications and changes may be made in the preferred and alternativeembodiments of the present invention without departing from its truespirit. For example, the boundary conditions of the multi-layer model ofFIG. 8 and TABLE 1 may be varied, the specific formulation of theanalytic solutions of TABLES 2-7 and other equations/formulas describedthroughout this paper may be adjusted or otherwise modified, thesimulators, couplings, and arrangement of the system may be selected toachieve the desired simulation. The simulations may be repeatedaccording to the various configurations, and the results compared and/oranalyzed.

This description is intended for purposes of illustration only andshould not be construed in a limiting sense. The scope of this inventionshould be determined only by the language of the claims that follow. Theterm “comprising” within the claims is intended to mean “including atleast” such that the recited listing of elements in a claim are an opengroup. “A,” “an” and other singular terms are intended to include theplural forms thereof unless specifically excluded.

1. A method of performing an oilfield operation of an oilfield having atleast one wellsite, each wellsite having a wellbore penetrating asubterranean formation for extracting fluid from an undergroundreservoir therein, the method comprising: obtaining a plurality ofreal-time parameters from a plurality of sensors disposed about theoilfield, wherein the plurality of real-time parameters comprise atleast one selected from a group consisting of real-time flow rate dataand real-time pressure data of the wellbore; configuring a gridlessanalytical simulator for simulating the underground reservoir based onthe plurality of real-time parameters; generating real-time simulationresults of the underground reservoir and the at least one wellsite inreal-time using the gridless analytical simulator; and performing theoilfield operation based on the real-time simulation results.
 2. Themethod of claim 1, wherein at least a portion of the oilfield is modeledas a vertically stacked system of a plurality of layers using aplurality of analytic solutions corresponding to the plurality oflayers, and wherein the gridless analytical simulator is based oncoupling the plurality of analytic solutions to account for crossflowamong the plurality of layers.
 3. The method of claim 2, wherein a fluxfield at an interface of the plurality of layers is obtained by solvinga Fredholm integral equation, and wherein a time evolution of the fluxfield is governed by a Volterra integral equation.
 4. The method ofclaim 1, wherein the oilfield comprises a plurality of wellsites, andwherein the gridless analytical model is configured to simulate aninterference effect from the plurality of wellsites.
 5. The method ofclaim 1, wherein the real-time simulation results are generated using atleast one selected from a group consisting of a no-flow boundarycondition, and a constant pressure boundary condition.
 6. The method ofclaim 1, wherein configuring the gridless analytical simulator comprisesidentifying a reservoir model based on at least one selected from agroup consisting of a neural network method, a rate of change of thereal-time pressure data, and a geological parameter.
 7. The method ofclaim 1, wherein the at least one wellsite comprises at least oneselected from a group consisting of a horizontal well, a vertical well,and a deviated well, and wherein the underground reservoir comprises aplurality of heterogeneous layers.
 8. The method of claim 1, wherein theunderground reservoir is a naturally fractured reservoir.
 9. The methodof claim 1, wherein hydraulic fracturing is performed for the at leastone wellsite.
 10. The method of claim 9, wherein the wellbore comprisesat least one selected from a group consisting of a finite conductivityhydraulic fracture and an infinite conductivity hydraulic fracture. 11.The method of claim 1, wherein the wellbore is modeled as a line sourcein the gridless analytical simulator.
 12. The method of claim 11,further comprising: simulating at least one selected from a groupconsisting of a wellbore storage effect and a finite wellbore radius byapplying corrections to the gridless analytical simulator.
 13. Themethod of claim 1, wherein the real-time simulation results comprises atleast one selected from a group consisting of reservoir pressure, flowrate, well skin, effective permeability, fracturing performance, welldrainage area, compartmentalization, and well productivity.
 14. Themethod of claim 1, wherein performing the oilfield operation comprisesat least one selected from a group consisting of anticipating an event,identifying an event, performing real-time diagnostics, performingreal-time interpretation, performing real-time decision making,performing real-time corrective action, and forecasting performance ofthe wellsite and the reservoir in real-time.
 15. The method of claim 1,further comprising: generating an alert based on comparing at least oneof the plurality of real-time parameters to a pre-determined limit; andclassifying the alert according to a plurality of pre-determined alertlevels, wherein an alert level of the plurality of pre-determined alertlevels dictates at least one selected from a group consisting of aproactive action and a reactive action.
 16. A method of performing anoilfield operation of an oilfield having a plurality of wellsites, eachwellsite having a wellbore penetrating a subterranean formation forextracting fluid from an underground reservoir therein, the methodcomprising: obtaining real-time pressure data from a permanent down-holepressure gauge; identifying a reservoir model for a gridless analyticalsimulator based on a rate of change of the real-time pressure data usinga neural network method; generating real-time simulation results of theunderground reservoir and the plurality of wellsites in real-time usingthe gridless analytical simulator; and performing the oilfield operationbased on the real-time simulation results.
 17. The method of claim 16,further comprising; filtering the real-time pressure data for at leastone selected from a group consisting of de-noising, outlier removal,transient identification, and data reduction.
 18. The method of claim16, further comprising; configuring the gridless analytical simulatorbased on a plurality of geological parameters obtained from a well log.19. The method of claim 16, further comprising; configuring the gridlessanalytical simulator based on a history matching process.
 20. The methodof claim 16, wherein the gridless analytical simulator is configured tosimulate an interference effect from the plurality of wellsites.
 21. Themethod of claim 16, further comprising: generating an alarm based oncomparing at least one of the real-time simulation results to apre-determined limit.
 22. The method of claim 21, wherein the alarmcomprises at least one selected from a group consisting of a drawn downalarm and a down time alarm.
 23. The method of claim 16, furthercomprising: updating the reservoir model based on comparing simulatedreal-time pressure data, obtained from the real-time simulation results,to the real-time pressure data obtained from the permanent down-holepressure gauge.
 24. The method of claim 16, wherein the real-timesimulation results comprises a trend of a wellbore skin, and whereinperforming the oilfield operation comprises scheduling a workoveroperation to reduce the wellbore skin.
 25. The method of claim 16,wherein the real-time simulation results comprises a trend of effectivepermeability, and wherein performing the oilfield operation comprisesdetermining a re-completion strategy.
 26. The method of claim 25,wherein the re-completion strategy comprises scheduling an artificiallift operation.
 27. A method of performing an oilfield operation of anoilfield having a plurality of gas wells, each gas well having awellbore penetrating a subterranean formation for extracting gas from anunderground reservoir therein, the method comprising: obtainingreal-time flow rate data from a flow meter; obtaining at least oneselected from a group consisting of real-time pressure data and offlinepressure data; generating a first simulation result of the undergroundreservoir and the plurality of gas wells using a non-linear regressionmodel with the real-time flow rate data, and the real-time pressuredata, and the offline pressure data if the real-time pressure data isnot available; identifying a reservoir model for a gridless analyticalsimulator using a neural network method if the real-time pressure datais available; generating a second simulation result of the reservoir andthe plurality of gas wells in real-time using the gridless analyticalsimulator; and performing the oilfield operation based on at least oneselected from a group consisting of the first simulation result and thesecond simulation result.
 28. The method of claim 27, furthercomprising; filtering the real-time flow rate data for at least oneselected from a group consisting of de-noising, outlier removal,transient identification, and data reduction.
 29. The method of claim27, further comprising; configuring the gridless analytical simulatorbased on a history matching process.
 30. A computer readable medium,embodying instructions executable by a computer to perform method stepsfor an oilfield operation, the oilfield having at least one wellsite,each of the at least one wellsite having a wellbore penetrating asubterranean formation for extracting fluid from an undergroundreservoir therein, the instructions comprising functionality to: obtaina plurality of real-time parameters from a plurality of sensors disposedabout the oilfield, wherein the plurality of real-time parameterscomprise at least one selected from a group consisting of flow rate andpressure of the wellbore; configure a gridless analytical simulator forsimulating the reservoir based on the plurality of real-time parameters;and generate real-time simulation results of the reservoir and the atleast one wellsite in real-time using the gridless analytical simulator,wherein the oilfield operation is performed based on the real-timesimulation results.
 31. A computer readable medium, embodyinginstructions executable by a computer to perform method steps for anoilfield operation, the oilfield having a plurality of wellsites, eachof the plurality of wellsites having a wellbore penetrating asubterranean formation for extracting fluid from an undergroundreservoir therein, the instructions comprising functionality to: obtainreal-time pressure data from a permanent down-hole pressure gauge;identify a reservoir model for a gridless analytical simulator based ona rate of change of the real-time pressure data using a neural networkmethod; generate real-time simulation results of the reservoir and theplurality of wellsites in real-time using the gridless analyticalsimulator; and perform the oilfield operation based on the real-timesimulation results.
 32. A computer readable medium, embodyinginstructions executable by a computer to perform method steps for anoilfield operation, the oilfield having a plurality of gas wells, eachof the plurality of gas wells having a wellbore penetrating asubterranean formation for extracting gas from an underground reservoirtherein, the instructions comprising functionality to: obtain real-timeflow rate data from a flow meter; obtain at least one selected from agroup consisting of real-time pressure data and offline pressure data;generate a first simulation result of the underground reservoir and theplurality of gas wells using a non-linear regression model with thereal-time flow rate data, and the real-time pressure data, and theoffline pressure data if the real-time pressure data is not available;identify a reservoir model for a gridless analytical simulator using aneural network method if the real-time pressure data is available;generate a second simulation result of the reservoir and the pluralityof gas wells in real-time using the gridless analytical simulator; andperform the oilfield operation based on at least one selected from agroup consisting of the first simulation result and the secondsimulation result.